# Efficacy of Bayesian Neural Networks in Active Learning

Vineeth Rakesh
Interdigital AI Lab
USA
<EMAIL_ADDRESS>Swayambhoo Jain
Interdigital AI Lab
USA
<EMAIL_ADDRESS>
###### Abstract

Obtaining labeled data for machine learning tasks can be prohibitively
expensive. Active learning mitigates this issue by exploring the unlabeled
data space and prioritizing the selection of data that can best improve the
model performance. A common approach to active learning is to pick a small
sample of data for which the model is most uncertain. In this paper, we
explore the efficacy of Bayesian neural networks for active learning, which
naturally models uncertainty by learning distribution over the weights of
neural networks. By performing a comprehensive set of experiments, we show
that Bayesian neural networks are more efficient than ensemble based
techniques in capturing uncertainty. Our findings also reveal some key
drawbacks of the ensemble techniques, which was recently shown to be more
effective than Monte Carlo dropouts.

##  1 Introduction

Although machine learning techniques have achieved a major breakthrough in
recent years, their performance comes at a cost of acquiring large volumes of
training data. This is especially true for supervised deep learning models
that demand a substantial amount of labeled data to achieve a reasonable
performance. For applications that require expert knowledge such as medical
and biological images, labels are extremely hard and expensive to obtain.
Active learning (AL) aims to mitigate this problem by smartly selecting data
points to label (from an expert) from a large pool of unlabeled data to
improve model performance. This sampling is typically based on some
acquisition function (AF) which provides a score for each unlabeled data that
signifies its level of importance. While there are many approaches to
implementing AF [1, 2], uncertainty-based based approaches are shown to be the
most effective [3, 4, 5, 6].

Bayesian neural network (BNN) naturally models uncertainty by learning a
probability distribution over the neural network weights. Therefore, for a
given input, as we take multiple realizations of the network, the variance
captured by the weights is reflected as the variation in the output, which in-
turn models uncertainty. BNNs learn by applying a prior distribution over
weights and performing variational inference to approximate the posterior
distribution. In [7], the authors proved that applying dropout to neural
networks is equivalent to a BNN. This theory was further leveraged by
proposing Monte Carlo Dropout (MCD) for uncertainty estimation in AL [5]. In a
recent work, [6] showed that ensemble of neural networks (EN) outperform MCD
when it comes to uncertainty estimation; thus, proving to be the choice for
active learning. Consequentially, it is natural to assume EN to perform better
than BNN since MCD is equivalent to BNN. However, dropout neural networks form
a special class of BNN where the posterior distribution is a special case of
_spike-and-slab_ distribution. Contrary to this, BNNs allow for a broader
class of prior and posterior distributions on weights.

In this paper, we re-establish the efficacy of BNNs in active learning over
ensembles and MCD by using a more general scaled normal prior based BNN
proposed in [8]. The scaled normal prior is a continuous relaxation of the
_spike-and-slab_ distribution and subsumes Dropout as a special case. Through
extensive experiments on multiple datasets namely, MNIST, Fashion MNIST,
CIFAR10 and CIFAR100 and a regression dataset on housing price prediction we
show that the scaled normal prior based BNN provides more robust and efficient
active learning over EN and MCD. We perform several experiments to demonstrate
the pros and cons of BNN over EN and MCD. For each round of active learning,
the models are trained using two different settings:  (1) re-use the trained
state of the model from previous round and retrain on the newly appended
datapoints (termed as continual training) and  (2) reset the model parameters
and retrain from scratch.  Our results show that BNN performs significantly
better than EN in terms of classification accuracy when it comes to continual
training. In fact, the performance of EN is worse than MCD which can be
attributed to overfitting and catastrophic forgetting. That being said, when
retrained from scratch, BNN and EN perform on a similar level, which is still
an advantage for BNN since estimating uncertainty using ensembles is a costly
process. We found that EN requires about five ensembles in order to achieve
good active learning performance. This implies, training of five different
i.i.d networks and storing the trained state of every single network instance.
BNN on other hand, achieves similar yet a more robust performance with a
trade-off of just doubling parameter size of conventional neural network.

Besides illustrating the overall effectiveness of BNN for active learning, we
answer the following questions:  (1) do acquisition functions behave the same
for Bayesian, ensemble and MC dropouts?  (2) how does model capacity affect
the outcome, do BNNs with lower model capacity perform worse than EN (or MCD)?
(3) are BNNs better than EN when predicting challenging class labels?
Inspired by the performance of BNNs, we also propose a computationally
efficient uncertainty estimation method for fully connected dense layers with
ReLU non-linearity. Since AL involves repeated uncertainty estimation over
large unlabeled dataset, efficient uncertainty estimation is of huge practical
importance. In the proposed method, instead of taking multiple instantiations
of neural networks to estimate the uncertainty, we perform just one forward
pass. In this forward pass, at each neuron, we approximate the probability
distribution parametrically. We show that this algorithm is capable of
achieving performances that is on-par with the traditional uncertainty
estimation in BNN.

To the best of our knowledge, we are the first to perform comprehensive
empirical analysis to demonstrate the efficacy of BNNs for active learning.
While most existing research limit themselves to experiments on small
architectures and dataset, ours does not have such constraints.

##  2 Related Work

Active learning has been explored extensively in classical machine learning
literature [9]. Much of the focus of classical literature has been on the high
dimensional data arising in the context of linear models such as support
vector machine [10, 11, 12]. That being said, recently, there has been a
significant interest in AL for deep neural networks (DNNs). While there are
many ways to perform AL in DNNs, uncertainty-based sampling techniques are the
preferred choice due to their ease of implementation and computational
efficiency. Uncertainty in the output of neural networks can be estimated
using (a) Bayesian neural networks or (b) ensemble of neural networks.

A popular technique for AL using BNNs is called Monte Carlo dropout which was
first proposed in [5]. The basic idea is to pass the new unlabeled data
through the DNNs multiple times while retaining the dropout layer. This
results in different realizations at the output of the DNNs. These
realizations can be used to estimate the uncertainty in output using various
measures such as entropy or the variance ratio. The estimated uncertainty can
then be utilized for acquiring new unlabeled data points thus performing
active learning.

On the other hand, ensemble based uncertainty estimation involves having an
ensemble of neural networks which typically have same architecture but trained
with different random initialization [4]. Uncertainty is estimated by passing
unlabeled examples through individual ensembles and their outputs are then
used to estimate the uncertainty. It was shown in [6] that ensemble methods
outperforms the Monte Carlo dropout-based estimation. This performance is
primarily attributed to the higher capacity and diversity in the ensemble
models as compared to different realizations of neural networks in dropout
based networks. While there are some studies that leverage BNNs for active
learning [4, 13, 14, 15], they have a few or all of the following
shortcomings:  (1) experiments are restricted to simple architectures with a
few dense or convolutional neural network (CNN) layers  (2) evaluation is
restricted to basic datasets such as MNIST  (3) no comparison of BNNs with
ensemble models  (4) claim to use BNNs, but actually use Monte Carlo dropouts
as approximation to BNNs.

##  3 Active Learning Via Bayesian Neural Networks

Bayesian neural network: For a given dataset
$\mathcal{D}=\\{(\mathbf{x}_{i},\mathbf{y}_{i})\\}_{i=1}^{D}$, Bayesian neural
networks involves calculation of the distribution of weights given the
training data $p(\mathbf{w}|\mathcal{D})$. The predictive distribution for a
test data $\mathbf{x}$ can be obtained by marginalizing $\mathbf{w}$ as
follows:
$p(\mathbf{y}|\mathbf{x},\mathcal{D})=\int{p(\mathbf{y}|\mathbf{x},\mathbf{w})p(\mathbf{w}|\mathcal{D})d\mathbf{w}}$.
This is equivalent to averaging predictions from an ensemble of neural
networks weighted by the posterior probabilities of their parameters
$\mathbf{w}$. However, exact computation of the posterior is intractable,
therefore, we resort to variational inference. That is, we wish to approximate
$p(\mathbf{w}|\mathcal{D})=p(\mathcal{D}|\mathbf{w})p(\mathbf{w})/p(\mathcal{D})$
by positing an approximate posterior $q_{\phi}(\mathbf{w})$ of variational
parameters $\phi$. The problem then reduces to optimizing the evidence-lower-
bound (ELBO) defined as follows:

$\displaystyle\mathcal{L}(\phi)=\underbrace{\mathbb{E}_{q_{\phi}(\mathbf{w})}[\log
p(\mathcal{D}|\mathbf{w})]}_{(a)}+\underbrace{\textrm{KL}[q_{\phi}(\mathbf{w})||p(\mathbf{w})]}_{(b)}$
(1)

where term (a) is the data-dependent likelihood term, and (b) is the
regularizer that measures the KL divergence between the posterior and prior.
For prior we use the scaled normal prior proposed in [8] in which the scales
$z$ follow a log-uniform prior: $p(z)\propto|z|^{-1}$. For a given layer
weight matrix $\mathbf{W}\in\mathbf{R}^{m\times n}$ for a fully connected
layer of neural network with input dimension $n$ and output dimension $m$ the
scales of shared across input dimension as

$\displaystyle
p(\mathbf{W},\mathbf{z})\propto\prod_{j=1}^{n}\frac{1}{|z_{j}|}\prod_{i,j}^{m,n}\mathcal{N}(w_{ij}|0,z_{j}^{2}).$
(2)

The main rationale behind scaled normal prior is that it is continuous
relaxation of the _spike-and-slab_ distribution. Dropout which is the basis of
MCD based active learning is a special case of _spike-and-slab_ distribution
[8] and we hope to get better performance with the scaled normal prior. We
consider the following joint approximate posterior

$\displaystyle
q_{\phi}(\mathbf{W},\mathbf{z})=\prod_{j}^{n}\mathcal{N}(z_{j}|\mu_{z_{j}},\sigma^{2}_{z_{j}})\prod_{i,j}^{m,n}\mathcal{N}(w_{ij}|z_{j}\mu_{ij},z_{j}^{2}\sigma_{ij}^{2}),$
(3)

and the corresponding ELBO is given by:

$\displaystyle\mathcal{L}(\phi)$
$\displaystyle=\mathbb{E}_{q_{\phi(\mathbf{z})}q_{\phi}(\mathbf{W}|\mathbf{z})}[\text{log}\,p(\mathcal{D}|\mathbf{W})]$
(4) $\displaystyle-$
$\displaystyle\mathbb{E}_{q_{\phi}(\mathbf{z})}[\textrm{KL}(q_{\phi}(\mathbf{W}|\mathbf{z}))||p(\mathbf{W}|\mathbf{z})]-\textrm{KL}(q_{\phi}(\mathbf{z})||p(\mathbf{z})).$

The KL divergences can be replaced with the closed form expressions

$\displaystyle\textrm{KL}(q_{\phi}(\mathbf{W}|\mathbf{z}))||p(\mathbf{W}|\mathbf{z})=\frac{1}{2}\sum_{i,j}^{m,n}\log\frac{e^{\sigma_{ij}^{2}+\mu_{ij}^{2}-1}}{\sigma_{ij}^{2}},$
$\displaystyle\textrm{KL}(q_{\phi}(z)||p(z))\approx\sum_{j}^{n}k_{1}\left(1-\gamma\left(k_{2}-k_{3}\alpha_{j}\right)-\frac{m(\alpha_{j})}{2k_{1}}\right),$

where $\alpha_{j}=-\log(\sigma_{z_{j}}^{2}/\mu_{z_{j}}^{2})$, $\gamma(\cdot)$
and $m(\cdot)$ are the sigmoid and soft-plus functions respectively and the
constants $k_{1}=0.63576,k_{2}=1.87320,k_{3}=1.48695$ [16].

From practical implementation point-of-view this scaled normal prior BNN is
easier to implement as compared to other prior. By virtue of the closed form
KL-divergences the ELBO in (4) can be optimized within the framework of
standard stochastic gradient ascent. In addition to this, during test time it
can implemented as a single feedforward pass where we replace $\mathbf{W}$ at
layer with its mean
$\tilde{\mathbf{W}}=\mathbf{M}_{W}\textrm{diag}(\mathbf{\mu}_{z})$ where
$\mathbf{M}_{W}$ is the matrix of means $\mathbf{\mu}_{ij}$ and
$\mathbf{\mu}_{z}$ is the vector of means $\mu_{z_{j}}$.

Acquisition functions: Once the model is trained on a small dataset, we use
acquisition functions (AF) to fetch the most uncertain datapoints. In a recent
work [6], the authors experimented with several acquisition functions and
found entropy [17] and variation-ratio to be the best candidates [18].
Therefore, we use these two metrics as our choice of AF. A BNN trained for
multi-class classification problem of $C$ classes maps a given data vector
$\mathbf{x}$ to a $C$ dimensional vector containing the probability of various
classes in its components. For a given unlabelled data vector $\mathbf{x}$ the
entropy is calculated by $T$ forward passes, each time with new weight
realization $\mathbf{W}_{t}$ from the trained posterior. First, the $T$
outputs vectors are averaged to obtain the probability for a given class $c$
as $\hat{p}(y=c|\mathbf{x})=1/T\sum_{t}p(y=c|\mathbf{x},\mathbf{W}_{t})$ where
$\mathbf{W}_{t}$ is the $t^{th}$ realization of weights obtained from the
trained posterior. Next, with these probability estimates the entropy can be
calculated as follows:

$\displaystyle
H(y|\mathbf{x})=-\sum_{c}\hat{p}(y=c|\mathbf{x})\log\left(\hat{p}(y=c|\mathbf{x})\right).$
(5)

The variation ratio can be calculated as $v=1-f_{m}/T$ where $f_{m}$ is the
number of predictions falling into the modal class category.

Active learning algorithm: With the Bayesian neural networks and acquisition
functions defined, we now describe our active learning methodology in
Algorithm 1. The procedure starts by training the model with some seed sample
$\mathcal{S}$ (line 1). The size of this sample could be anywhere between 2-5
percent of the training set (depending on the complexity of data), we call
this step as seed training. At each round $r\in R$, we add $k$ new samples by
calling the active learning functions (line 1). The function first removes the
chosen sample $\mathcal{S}$ from the main dataset $\mathcal{D}$ and creates
$T$ instances of networks by sampling weights that was learned during the
training phase. Each instance is tested on unseen data points to obtain
ensemble of outputs (lines 1-1). Depending on the type of AF (i.e., variation-
ratio or entropy), the uncertainty over ensembles is calculated and the new
data-points are chosen based on the largest uncertainty score. Once the model
is trained on newly appended data points, the algorithm proceeds by validating
on the held-out training dataset $\mathcal{D}_{v}$ (line 1). Note that it is
not necessary to use the validation data $D_{v}$ and it is entirely optional.
We assumed that even in active learning setting, we can afford using a very
small percentage of unseen data for validation. In the subsequent rounds, the
best weights from the previous round are loaded (i.e., based on the validation
set) and the algorithm resumes performing AL and re-training. Unlike [6],
after each round we do not retrain the model from scratch. In our experiments,
we found that retraining from scratch does not perform as well as re-using
weights from previous rounds and re-training.

Accelerating uncertainty estimation: As discussed earlier while the scaled
normal prior BNN allows inference in a single feedforward pass but for the
uncertainty estimation passing it still requires a given unlabelled example
passed through multiple realization weights using the trained posterior. This
is computationally expensive and we alleviate this by approximating the
probability distribution of the input to the last layer directly. Equipped
with this distribution we propose to directly generate the random realizations
of the input before last layer non-linearity and use those to estimate the
uncertainty.

For the scaled normal posterior in (3) the analytical expression for the
distribution is challenging due to the non-linear transformations in the
neural network and dependencies due to structure of layers such as convolution
layers. To illustrate this, consider a simple linear transformation of
$\mathbf{x}$ by matrix $\mathbf{W}\in\mathbb{R}^{m\times n}$, bias vector
$\mathbf{b}\in\mathbb{R}^{m}$ given by $\mathbf{y}=\mathbf{Wx}+\mathbf{b}$
where $\mathbf{W}$ follows the posterior in (3). Each entry of $\mathbf{y}$ is
a sum of $n$ independent random variables with scaled normal distribution as
$y_{i}=\sum_{j=1}^{n}W_{ij}x_{j}+b_{i}$. Considering the fact that a typical
neural network has multiple layers and non-linearities, the calculation of
analytical distribution is non-trivial. However, under the assumption that
$y_{i}$ is Gaussian, the mean and variance of $y_{i}$ after passing through
ReLU non-linearity is analytically tractable. Based on this, for a multi-layer
neural network comprising of $L$ dense layers we can obtain the distribution
of components vectors before the non-linearity from the distribution of layer
input. Suppose weights of $l^{th}$ layer are represented by
$\mathbf{W}^{l}\in\mathbb{R}^{n^{l}\times n^{l+1}}$ where $n^{l+1},n^{l}$ are
the input and output dimension of this layer. Then the expectation and
variance of components of the vector
$\mathbf{y}^{l}=\mathbf{W}^{l}\mathbf{x}^{l-1}+\mathbf{b}^{l}$ can be obtained
terms of first and second order moments of components of $\mathbf{x}^{l-1}$ as
follows

$\displaystyle\mathbb{E}\left[y_{i}^{l}\right]$
$\displaystyle=\sum_{j=1}^{n^{l}}\mathbb{E}\left[w^{l}_{ij}\right]\mathbb{E}\left[x^{l-1}_{j}\right]+b_{i}^{l},$
(6) $\displaystyle\mathbb{V}[y_{i}^{l}]$
$\displaystyle=\sum_{j=1}^{n^{l}}\mathbb{E}\left[\left(w^{l}_{ij}\right)^{2}\right]\mathbb{E}\left[\left(x^{l-1}_{j}\right)^{2}\right]$
$\displaystyle\quad\quad\quad-\mathbb{E}\left[x^{l-1}_{j}\right]^{2}\mathbb{E}\left[w^{l}_{ij}\right]^{2},$
(7)

where
$\mathbb{E}\left[w^{l}_{ij}\right]^{2}=(\mathbf{\sigma}_{z_{j}}^{2}+\mu_{z_{j}}^{2})(\sigma_{ij}^{2}+\mu_{ij}^{2})$.
Further, the input to next layer is obtained passing $\mathbf{y}^{l}$ through
a ReLU non-linearity as
$\mathbf{x}^{l}=\textrm{ReLU}\left(\mathbf{y}^{l}\right)$. We assume that non-
linearity until the last layer is ReLU. Finally, assuming that components of
$\mathbf{y}^{l}$ are Gaussian with mean and variance computed as above, the
mean and variance of components of $\mathbf{x}^{l}$ are given below

$\displaystyle\mathbb{E}\left[x_{i}^{l}\right]$
$\displaystyle=\mathbb{E}\left[y_{i}^{l}\right]\Phi(\delta_{i}^{l})+\mathbb{V}[y_{i}^{l}]f(-\delta_{i}^{l}),$
(8) $\displaystyle\mathbb{E}[(x_{i}^{l})^{2}]$
$\displaystyle=\left(\mathbb{E}\left[y_{i}^{l}\right]^{2}+\mathbb{V}\left[y_{i}^{l}\right]\right)\Phi(\delta_{i}^{l})$
$\displaystyle\quad+\sqrt{\mathbb{E}\left[y_{i}^{l}\right]^{2}\mathbb{V}\left[y_{i}^{l}\right]}f(\delta_{i}^{l})$
(9)

where
$\delta_{i}^{l}=\mathbb{E}\left[y_{i}^{l}\right]/\mathbb{V}\left[y_{i}^{l}\right]$
, $\Phi$ and $f$ are c.d.f. and p.d.f. of standard Gaussian distribution.
Using above equations (6) to (9) the mean and variance at the input before
non-linearity of each layer can be computed iteratively till the last layer
$\mathbf{y}^{L}$. Equipped with this information the uncertainty can then be
computed by directly generating $\mathbf{y}^{L}$ and then passing them through
the last layer’s non-linearity.

1 Inputs: Training dataset $\mathcal{D}$, validation dataset
$\mathcal{D}_{v}$, test dataset $\mathcal{D}_{t}$, seed dataset $\mathcal{S}$,
model $\mathcal{M}$, number of epochs $E$, mini batch size $b$, number of
rounds $R$, acquisition size $k$, number of instances $T$.

2 ActiveSubSelectData  _(typ)_

3 $\mathcal{D}\leftarrow\mathcal{D}-\mathcal{S}$

4 $ENO\leftarrow[\,]$(array to hold ensemble outputs)

5 for  _$j\in T$_ do

6 $\mathcal{M}_{s}\sim\mathcal{M}(\mathbf{w})$(sample from a weight instance)

7 $ENO\leftarrow$ append the model output $\mathcal{M}_{s}(\mathcal{D})$

8 $new\leftarrow$ get uncertainty of ENO with eq(5) if $typ$ is ”Entropy” else
use variation-ratio $v$

9 $new\leftarrow$ sort $new$ and get top-$k$ datapoints with the most
uncertainty

10 $\mathcal{S}\leftarrow$ $\mathcal{S}\cup new$

11

12

13Train($\mathcal{M}$) with seed sample

14 for  _each $r\in R$_ do

15 ActiveSubSelectData(”Entropy/Variation-Ratio”)

16 for  _each $e\in E$_ do

17

18 Train($\mathcal{M}$) on $\mathcal{S}$

19 $v\\_loss\leftarrow$ Evaluate($\mathcal{M}$) on $\mathcal{D}_{v}$ and get
validation loss

20 if EarlyStoppingCriteria($v\\_loss$): break

21 $\mathcal{M}\leftarrow$ Load best weights based on $\mathcal{D}_{v}$ (if
continual training)

Test($\mathcal{M}$) on $\mathcal{D}_{t}$

Algorithm 1 Uncertainty-based smart data sampling

##  4 Experiments

Starting from simple multi-layer perceptrons to deep CNN models we perform
several experiments to understand the effectiveness and shortcomings of BNNs
over EN and MCD. Our objective is to analyze BNNs from the following
perspective:  (1) overall efficiency in acquisition of new data points  (2)
robustness of BNNs during minimal retraining (i.e., retraining by reusing the
trained model from previous round) and  (3) impact of model capacity, and
ensemble size.  In addition to this, we also show the outcome of the proposed
accelerated uncertainty estimation over dense neural networks.

Dataset and models: The experiments are performed on four image classification
datasets and one regression dataset. For classification, we chose MNIST,
Fashion MNIST (FMNIST) [19], CIFAR10 and CIFAR100 [20]. For regression, we
chose the housing price prediction dataset introduced by [21]. It consists of
535 unique houses sampled from the state of California. Each house is
represented by both visual and textual data with the visual features
representing the front side of the house, the kitchen, the bedroom and the
bathroom. The following neural network architectures are used in this paper
(for detailed description please refer Appendix 6.1).  (a) LeNetD2: a simple
densely connected network with 300 and 100 neurons in the first and second
layer respectively,  (b) LeNet5[22]: consists of 2 CNN layers followed by a
classifier network with three dense layers,  (c) AlexNet Light (ANL): this is
simplified version of the Alexnet architecture that is similar to the K-CNN
used in [6]. It has 4 CNN layers followed by two dense layers,  (d) VGG: there
are several variations of VGG. For our experiments, we use VGG19 [23] with 16
CNN layers followed by the classifier network, and  (e) Densenet: we use
Densenet121 [24] with a growth rate of 32.  The Bayesian versions of these
models were implemented from scratch. At the time of writing, there are no
standardized module for Pytorch111 Models were implemented using
Pytorch(pytorch.org) and Numpy(numpy.org) libraries that can seamlessly
convert a conventional neural network to its Bayesian counterpart. The
complete active learning library consisting of all the models and experiments
performed in this paper will be publicly hosted via Github
222github.com/VRM1/ActiveLearning.

Active learning setting: Our training procedure was explained in Algorithm 1.
Depending on the model complexity and the dataset, the number of rounds $R$ is
set anywhere between $40$ to $80$. The number of data-points added during each
round (i.e, $k$) varies from $5$ to $250$. The number of neural network
instances (NNI) to compute uncertainty (i.e., the variable $T$) is set as $15$
for Densenet and $25$ for all other neural networks. For BNN, the instances
are created by sampling from the posterior distribution of weights and for
MCD, the dropouts are activated during the active learning phase (but turned
off during testing phase). Similar to [6], for EN, the number of NNI are set
as 5 and the weights are initialized using the default Pytorch setting, which
follows Kaiming uniform distribution. The details of neural network
architecture, dataset, epochs, acquisition size, etc., are summarized in Table
1. Experiments are performed by either reusing the state of the model from
previous round and retraining, termed as continual training (CT) or completely
resetting the model and retraining from scratch (RFS). In CT, in each new
round, models are trained for a significantly lower number of epoch, typically
around 30-50, depending on the type of model and dataset.On the contrary, in
RFS the number of epochs range anywhere between 100-200, which makes the
training time much greater than CT.

Evaluation metrics: For classification task, the following metrics are used
(1) Top-1 Accuracy: is the ratio between the number of correct predictions and
the total number of predictions. We take the class that corresponds to the
highest probability (i.e., from the softmax layer) as the predicted label,
(2) Pr: precision is the fraction of true positives among the retrieved
instances, and  (3) F1: F1 is the harmonic mean of the precision and recall,
where recall is defined as the fraction of the total relevant instances (i.e.,
both true positives and true negatives) that were retrieved.

Aside from the aforementioned performance metrics, it is also important to
know the calibration of our models since a high accuracy does not imply good
calibration and vice versa [25]. Therefore, for classification tasks, we use
expected calibration error (ECE) [26] as the metric of choice. ECE
approximates the level of calibration by partitioning predictions into $M$
equally-spaced bins and taking a weighted average of the bin’s
accuracy/confidence difference. More precisely,

$\displaystyle ECE=\sum_{m=1}^{M}\cfrac{|B_{m}|}{n}|acc(B_{m})-conf(B_{m})|$
(10)

where n is the total number of samples and $B_{m}$ indicates a group of
samples whose prediction confidence falls into a certain interval.
$acc(B_{m})$ is the ground truth accuracy and $conf(B_{m})$ is predicted
probability (termed as confidence) of $B_{m}$. For regression task, we use the
the coefficient of determination $R^{2}$, which is a measure of the closeness
of the predicted model relative to the actual model [27]. $R^{2}$ is defined
as $1-SSE/SST$, where $SSE=\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}$ and
$SST=\sum_{i=1}^{n}(\bar{y}-y_{i})^{2}$, $\hat{y}_{i}$ is the predicted value
of $i$ and $\bar{y}$ is the observed average.

Dataset | Model | Epoch(CT/RFS) | #Rnd | Aq Size
---|---|---|---|---
F/MNIST | LenetD2 | 30/100 | 40 | 1K/100/4K
F/MNIST | Lenet5 | 30/100 | 40 | 100/100/4K
CIFAR10 | VGG16 | 50/200 | 80 | 1K/250/20K
CIFAR100 | VGG16 | 50/200 | 80 | 1K/250/20K
Housing | LenetD2 | 50 | 40 | 50/5/200
Table 1: Settings of active learning experiments for various datasets. The
acquisition size (Aq Size)is the number of new datapoints added in each round
and #Rnd is the number of rounds. Aq Size is divided into 3 parts (separated
by /), where the first indicates seed sample size, the second is the #
datapoints added in each round and the third is the total aquisition size
across all rounds.

###  4.1 Results

(a) (a)

(b) (b)

(c) (d)

(d) (c)

Figure 1: Active learning performance on LeNetD2 and Lenet5 architecture with
CT setting. For MNIST, BNN and EN perform on a similar level, while MCD trails
behind. For FMNIST, BNN clearly outperforms the rest and although EN performs
similar to MCD upto round 10, it clearly starts to trail behind EN and
surprisingly even looses to MCD.

Performance on shallow neural networks: We begin by looking at the accuracy
performance of BNN and other baselines over LeNetD2 and Lenet5 architectures.
In Figure 1, x-axis is the number of samples at each round, which is indicated
as $(\times 100)$. For instance at $x=5$, we have added 500 samples and round
0 marks the beginning of the active learning procedure. Each model is followed
by a hyphen and a letter, which indicates the type of acquisition function.
For example, E is entropy and VR is variation ratio. For representation
purpose we do not show random sampling, but in our experiments they
substantially trailed behind every aquisition function. Overall, across all
models, VR tends to outperform entropy when it comes to acquisition functions.
Nonetheless, this difference is more pronounced during the first half of
rounds, during the final rounds the difference becomes quite narrow. This
shows that EN tends to improve its acquisition quality when it sees more data.
When it comes to MNIST, BNN and EN perform on a similar level, while MCD
trails behind. However, due to the simplicity of dataset, the difference in
performance is not that discernible. For FMNIST, which is a more challenging
dataset, we start to see some interesting differences. In LenetD2 (Figure 1
(b)), while the performance of BNN and EN are on-par with each other (for VR),
BNN seems to yield better results for entropy. In Lenet5 (Figure 1 (d))
although EN performs similar to BNN upto round 10, BNN clearly produces better
accuracy than all other models upto round 50. Additionally, we observe EN’s
accuracy lagging quite significantly behind the rest after about 30 rounds.
This was quite surprising to us since in [6], the authors claim EN to perform
better than MCD. Upon further investigation, we found that the poor
performance of EN (compared to MCD) is only observed with CT. In the upcoming
experiments, we will show the results on both CT and RFS.

(a) (a)

(b) (b)

(c) (e)

(d) (f)

(e) (c)

(f) (d)

(g) (g)

(h) (h)

Figure 2: Active learning performance of ANL and VGG over Cifar10 –
classification accuracy and model calibration. BNN yields a more robust
performance when compared to EN and MCD, irrespective of the training
methology (a)-(d), while EN suffers quite significantly in the CT setting.
When retraining from scratch, EN gets a major boost in performance and just
slightly lags behind BNN for ANL and performs on par with BNN on VGG.
Unfortunately, BNN suffers from poor calibration (e)-(h), but it can be quite
easily corrected. The calibrated model BNN-VRc is shown as the light blue bar.

Performance on Cifar10: The active learning performance of ANL and VGG is
represented in Figure 2 (a)-(d). The results unravel some important
characteristics of the models. First, when it comes to CT, BNNs performance is
significantly better than both EN and MCD, especially for VGG. Here, BNN
achieves about 75% and MCD about 67% accuracy, but similar to previous results
(i.e., Lenet5) EN clearly under performs with just 55%. A possible reason for
this outcome could be the lack of regularization. ENs are full capacity models
without any dropouts and we observe quite substantial overfitting during CT
setting. This in-turn result in making incorrect decisions when acquiring new
data points during the active learning phase. Another reason could be
catastrophic forgetting, which is a well known problem in continual training
of neural networks [28, 29]. BNN on the other hand seems to be more robust to
such perturbations. That being said, when retraining from scratch EN starts to
outperform MCD as it doesn’t overfit the data acquired in previous arounds,
which aligns with the recent study [6]. For ANL (Figure 2 (b)) we clearly see
EN outperforming MCD, however, BNN still achieves better performance. When it
comes to VGG (Figure 2 (c)) we don’t not find any distinct performance gaps
between all three models. Overall, when retrained from scratch, all models
perform better compared to CT, but we observe a lot more variations in the
accuracy scores, while in CT the increase in accuracy is quite smooth.

Calibration characteristics: When it comes to measuring robustness of machine
learning models, relying solely on performance metrics such as accuracy is not
sufficient. In our experiments, the calibration score is measured as expected
calibration error (ECE) using equation (10). In Figure 2 (e)-(h), the y-axis
is the ECE and the x-axis is the elapsed round number. For instance, for both
ANL and VGG since the active learning is performed until round 80 (Figure 2
(a)-(d)), 25% implies the 20th round of the corresponding experiment. From
calibration plots, we observe that ENs have the best out-of-the-box ECE scores
followed by MCD. Although BNN offers great performance in terms of accuracy,
they seem to be quite poorly calibrated. Thankfully, calibration is a post-
training step and there are several ways to calibrate a model such as
histogram binning and temperature scaling; in this paper we adopt the later.
The ECE of the calibrated BNN model is shown as the blue bar. It is important
to note that even though we have significantly reduced the calibration error,
the accuracy of BNN still remains unaltered.

(a) (a)

(b) (b)

Figure 3: Active learning performance of Densenets on Cifar100 dataset.
Similar to the results on Cifar10, BNN produces a more robust performance than
EN and MCD whether it is CT or retraining from scratch.

Performance on Cifar100: From Figure 3, one can observe that even for larger
neural networks such as Densenets, BNN produces significantly better accuracy
compared to MCD and EN. As for EN, it looks like as the neural networks become
more complex, the CT methodology starts to substantially cripple the
performance. When RFS both BNN and EN have similar performance, while MCD
trails behind.

Performance on regression dataset: The robust performance of BNN is not just
restricted to classification, but also extends to regression. Figure 4 shows
the performance of BNN in terms of $R^{2}$ on housing price prediction. For
this dataset, we used the LenetD2 architecture where the input layer is a
feature vector that is from by concatenating the image and descriptive text
features of the individual houses [21]. For this dataset, we start with a seed
sample of 50 and in each round we add five new data points that is selected
via active learning. Here there is no major performance difference between BNN
and EN, but MCD clearly under performs. Additionally, when it comes to BNN
there seems to be a lot more variance in $R^{2}$ score in each round, while EN
tends to be smoother.

Figure 4: Housing price prediction dataset using LenetD2. There is no major
performance difference between BNN and EN, but MCD clearly under performs.

###  4.2 Ablation Study

| Class | BNN-VR | EN-VR
---|---|---|---
| 25% | 50% | 100% | 25% | 50% | 100%
Pr | F1 | Pr | F1 | Pr | F1 | Pr | F1 | Pr | F1 | Pr | F1
Cifar10 (VGG) | Bird | 51.2 | 47.9 | 63.8 | 67.1 | 79.3 | 76.9 | 48.8 | 46.2 | 62.8 | 63.4 | 75.7 | 75.2
Cat | 50.6 | 45.4 | 59.6 | 57.8 | 67.3 | 68.4 | 49.3 | 44.9 | 58.4 | 55.6 | 69 | 68.2
Dog | 36.4 | 44.0 | 62.7 | 65.1 | 77.8 | 77.2 | 54.1 | 58.2 | 60.9 | 64.9 | 74 | 74.9
FMNIST (Lenet5) | Pullover | 74.7 | 75.3 | 81 | 79.8 | 81.8 | 80.2 | 71.4 | 72.4 | 74.4 | 75.2 | 78.3 | 77
Shirt | 57.1 | 62.8 | 62.7 | 64.5 | 66 | 66.3 | 56.6 | 56.2 | 61.9 | 60.5 | 62.7 | 63.4

Table 2: Precision and F1 score of BNN and EN on challenging class labels. The
second row indicates the elapsed round and Pr indicates precision. Even though
for VGG, EN and BNN have similar accuracy (Figure 2(d)), the precision and F1
measure on challenging labels indicate that performing AL via BNN leads to
better results.

We perform data-centric and model-centric ablation studies to get a deeper
understanding of BNN’s performance.

Performance on challenging class labels: When retraining from scratch, we see
that the performance gap between BNN and EN is quite narrow, especially on VGG
(Figure 2(d)). Nonetheless, accuracy is a global score and when it comes to
AL, it is important to measure the performance on labels that are hard to
classify. Table 2 shows the precision and F1 measure of FMNIST and Cifar10
datasets for classes that are most challenging to predict. The corresponding
accuracy plots can be seen from Figure 1(d) and Figure 2(d) respectively. Even
though BNN and EN produce similar accuracy, for more challenging class labels,
the former outshines the later. Across all selected rounds of active learning,
both precision and F1 measures of BNN are clearly better than EN. This shows
that BNN’s uncertainty estimation is more effective in acquiring datapoints
that improve the model performance on challenging class labels.

(a) (a)

(b) (b)

Figure 5: Impact of number of NNI and model capacity on Cifar10. BNN-VR25
indicates 25 network instances used to estimate the uncertainty, while BNN-VR5
implies five. BNN-VR- is the ANL network with reduced capacity that matches
those of MCD when dropouts are activated.

Impact of NNI: The performance of Al is based on how robust is the uncertainty
estimation. When it comes to BNN, a key factor that determines the goodness of
estimation is the number of NNI (i.e., lines 5-7 of Algorithm 1). A major
advantage of EN over BNN (and MCD) is the low number of NNI that is needed to
estimate uncertainty, which is set as five in our experiments. On the other
hand, for BNNs we have 25 instances. Therefore, we wanted to test how good is
BNN when the NNI is reduced to that of EN. The results of this experiment is
shown in Figure 5. Here, one can observe that during CT, there is some
performance dip but it is mostly towards the final rounds. On the other hand,
for RFS, there is no noticeable loss in accuracy. We observed similar results
even with entropy as the acquisition function. A possible reason for this
outcome could be attributed to Bayesian’s ability to learn distribution over
weights (instead of point estimate), which naturally models uncertainty during
the training process (which EN and MCD lacks).

Impact of reduced model capacity: Besides NNI, another factor that determine
the performance of AL is the model capacity. Both BCN and EN enjoy the benefit
of being a full capacity model (during training). However, for MCD, due to
dropouts a significant portion of neurons in dense and CNN layers remain
inactive, which is one of main reasons for MCD’s inferior performance. In
fact, [6] show that when ENs are capacity-limited, their performance drops
roughly to that of MCD. To see this effect on BCN, we reduced the number of
CNN filters and dense layers to that of MCD (i.e., 50% less for dense and 25%
less for CNN). The outcome of this experiment is shown in Figure 5, where BNN-
VR- is the capacity-limited BCN. While we do see a noticeable performance dip
when compared to the regular BNN, the accuracy still manages to be on-par with
EN for both CT and RFS.

Performance of accelerated uncertainty estimation: Finally, we compare the
performance of the proposed Bayesian accelerated uncertainty estimation
learning (AUE) on LenetD2 architecture in Figure 6. It is interesting to see
that for MNIST, our approximation is as effective as uncertainties estimated
through iterative realizations of neural networks. For FMNIST (which is a more
challenging dataset), AUE still produces respectable results. It is important
to note that iterative procedure creates multiple instantiations of the
network and every instance needs to see the entire dataset to estimate
uncertainty. Although, AUE cannot match the performance of the iterative
approach, it significantly reduces the time taken to calculate the uncertainty
and this might be a worthy trade-off between performance and speed.

(a) (a) (b) (b)

Figure 6: Active learning performance of iterative uncertainty estimation Vs
accelerated uncertainty estimation (BNN-AUE) on MNIST and FMNIST datasets.

##  5 Conclusion

In this paper, we analyzed Bayesian neural networks for active learning and
showed that they are more efficient than ensemble and Monte Carlo dropouts in
capturing uncertainty. Our experiments revealed the following characteristics
of BNN:  (a) BNNs are more robust when it comes to continual learning, and EN
performs worse than MCD in this setting,  (b) produces respectable uncertainty
estimates even with reduced model capacity, and  (c) produces better precision
and F1 on difficult class labels compared to EN.  To overcome the computation
cost of repeated uncertainty estimation in active learning, we proposed an
accelerated uncertainty estimation technique for dense layers.

## References

  * [1] Ozan Sener and Silvio Savarese.  A geometric approach to active learning for convolutional neural networks.  arXiv preprint arXiv, 1708:1, 2017.
  * [2] Lin Yang, Yizhe Zhang, Jianxu Chen, Siyuan Zhang, and Danny Z Chen.  Suggestive annotation: A deep active learning framework for biomedical image segmentation.  In International conference on medical image computing and computer-assisted intervention, pages 399–407. Springer, 2017.
  * [3] Yi Yang, Zhigang Ma, Feiping Nie, Xiaojun Chang, and Alexander G Hauptmann.  Multi-class active learning by uncertainty sampling with diversity maximization.  International Journal of Computer Vision, 113(2):113–127, 2015\.
  * [4] Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell.  Simple and scalable predictive uncertainty estimation using deep ensembles.  In Advances in neural information processing systems, pages 6402–6413, 2017.
  * [5] Yarin Gal, Riashat Islam, and Zoubin Ghahramani.  Deep bayesian active learning with image data.  In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1183–1192. JMLR. org, 2017.
  * [6] William H Beluch, Tim Genewein, Andreas Nürnberger, and Jan M Köhler.  The power of ensembles for active learning in image classification.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9368–9377, 2018.
  * [7] Yarin Gal and Zoubin Ghahramani.  Dropout as a bayesian approximation: Representing model uncertainty in deep learning.  In international conference on machine learning, pages 1050–1059, 2016.
  * [8] Christos Louizos, Karen Ullrich, and Max Welling.  Bayesian compression for deep learning.  In Advances in Neural Information Processing Systems, pages 3288–3298, 2017.
  * [9] Burr Settles.  Active learning literature survey.  Technical report, University of Wisconsin-Madison Department of Computer Sciences, 2009.
  * [10] Alex Holub, Pietro Perona, and Michael C Burl.  Entropy-based active learning for object recognition.  In 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pages 1–8. IEEE, 2008.
  * [11] Ajay J Joshi, Fatih Porikli, and Nikolaos Papanikolopoulos.  Multi-class active learning for image classification.  In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 2372–2379. IEEE, 2009.
  * [12] Xin Li and Yuhong Guo.  Adaptive active learning for image classification.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 859–866, 2013.
  * [13] Andreas Kirsch, Joost van Amersfoort, and Yarin Gal.  Batchbald: Efficient and diverse batch acquisition for deep bayesian active learning.  In Advances in Neural Information Processing Systems, pages 7026–7037, 2019.
  * [14] Parmida Atighehchian, Frédéric Branchaud-Charron, and Alexandre Lacoste.  Bayesian active learning for production, a systematic study and a reusable library.  arXiv preprint arXiv:2006.09916, 2020.
  * [15] Prateek Munjal, Nasir Hayat, Munawar Hayat, Jamshid Sourati, and Shadab Khan.  Towards robust and reproducible active learning using neural networks.  arXiv, pages arXiv–2002, 2020.
  * [16] Dmitry Molchanov, Arsenii Ashukha, and Dmitry Vetrov.  Variational dropout sparsifies deep neural networks.  In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 2498–2507. JMLR. org, 2017.
  * [17] Claude E Shannon.  A mathematical theory of communication.  Bell system technical journal, 27(3):379–423, 1948.
  * [18] Linton C Freeman.  Elementary applied statistics: for students in behavioral science.  John Wiley & Sons, 1965.
  * [19] Han Xiao, Kashif Rasul, and Roland Vollgraf.  Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms.  arXiv preprint arXiv:1708.07747, 2017.
  * [20] Alex Krizhevsky, Geoffrey Hinton, et al.  Learning multiple layers of features from tiny images.  2009\.
  * [21] Eman H Ahmed and Mohamed Moustafa.  House price estimation from visual and textual features.(2016).  arXiv preprint cs.CV/1609.08399, 2016.
  * [22] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner.  Gradient-based learning applied to document recognition.  Proceedings of the IEEE, 86(11):2278–2324, 1998.
  * [23] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [24] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger.  Densely connected convolutional networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.
  * [25] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger.  On calibration of modern neural networks.  arXiv preprint arXiv:1706.04599, 2017.
  * [26] Mahdi Pakdaman Naeini, Gregory F Cooper, and Milos Hauskrecht.  Obtaining well calibrated probabilities using bayesian binning.  In Proceedings of the… AAAI Conference on Artificial Intelligence. AAAI Conference on Artificial Intelligence, volume 2015, page 2901\. NIH Public Access, 2015.
  * [27] Nico JD Nagelkerke et al.  A note on a general definition of the coefficient of determination.  Biometrika, 78(3):691–692, 1991.
  * [28] Ian J Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, and Yoshua Bengio.  An empirical investigation of catastrophic forgetting in gradient-based neural networks.  arXiv preprint arXiv:1312.6211, 2013.
  * [29] Ronald Kemker, Marc McClure, Angelina Abitino, Tyler Hayes, and Christopher Kanan.  Measuring catastrophic forgetting in neural networks.  arXiv preprint arXiv:1708.02072, 2017.


# On Strong Small Loop Transfer Spaces Relative to Subgroups of Fundamental
Groups

S.Z. Pashaei<EMAIL_ADDRESS>B. Mashayekhy<EMAIL_ADDRESS>M. Abdullahi Rashid<EMAIL_ADDRESS>Department of Pure Mathematics,
Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University
of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran.

###### Abstract

Let $H$ be a subgroup of the fundamental group $\pi_{1}(X,x_{0})$. By
extending the concept of strong SLT space to a relative version with respect
to $H$, strong $H$-SLT space, first, we investigate the existence of a
covering map for strong $H$-SLT spaces. Moreover, we show that a semicovering
map is a covering map in the presence of strong $H$-SLT property. Second, we
present conditions under which the whisker topology agrees with the lasso
topology on $\widetilde{X}_{H}$. Also, we study the relationship between open
subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$. Finally, we
give some examples to justify the definition and study of strong $H$-SLT
spaces.

###### keywords:

Strong small loop transfer space, quasitopological fundamental group, whisker
topology, lasso topology, covering map, semicovering map.

###### MSC:

[2010]57M10, 57M12, 57M05, 55Q05.

##  1 Introduction and Motivation

Throughout this article, we consider a path connected topological space $X$
with a base point $x_{0}\in{X}$. Given a pointed topological space
$(X,x_{0})$, we denote the set of all paths in $X$ starting at $x_{0}$ by
$P(X,x_{0})$. Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$ and
$\widetilde{X}_{H}=P(X,x_{0})/\sim$, where $\alpha\sim\beta$ if and only if
$\alpha(1)=\beta(1)$ and $[\alpha\ast\beta^{-1}]\in{H}$. The equivalence class
of $\alpha$ is denoted by $[\alpha]_{H}$. We denote the constant path at
$x_{0}$ by $c_{x_{0}}$. Note that $[\alpha]_{H}=[c_{x_{0}}]_{H}$ if and only
if $[\alpha]\in{H}$. The map $p_{H}:\widetilde{X}_{H}\rightarrow X$ is defined
to be the endpoint projection $p_{H}([\alpha]_{H})=\alpha(1)$. We denote
$p_{e}$ and $\widetilde{X}_{e}$ instead of $p_{H}$ and $\widetilde{X}_{H}$,
respectively, when $H$ is the trivial subgroup.

There are three famous topologies on $\widetilde{X}_{H}$. One of them is the
quotient topology induced by the compact-open topology on $P(X,x_{0})$. We
denote the space $\widetilde{X}_{H}$ equipped with this topology by
$\widetilde{X}^{top}_{H}$. The second topology is the whisker topology which
was introduced by Spanier [14, Theorem 2.5.13] and named by Brodskiy et al.
[6], as follows.

######  Definition 1.1.

For any pointed topological space $(X,x_{0})$ the whisker topology on the set
$\widetilde{X}_{H}$ is defined by the collection of all the following sets as
a basis

$B_{H}([\alpha]_{H},U)=\\{[\beta]_{H}\in{\widetilde{X}_{H}}\ |\
\beta\simeq\alpha\ast\lambda\ for\ some\ \lambda:I\rightarrow U,\
\lambda(0)=\alpha(1)\\},$

where $[\alpha]_{H}\in{\widetilde{X}_{H}}$ and $U$ is an open neighborhood of
$\alpha(1)$. In the case $H=1$, we denote the basis elements of
$\widetilde{X}_{e}$ by $B([\alpha],U)$.

The Spanier group $\pi(\mathcal{U},x)$ [14] with respect to an open cover
$\mathcal{U}=\\{U_{i}\ |\ i\in{I}\\}$ is defined to be the subgroup of
$\pi_{1}(X,x)$ which contains all homotopy classes having representatives of
the type $\prod_{j=1}^{n}\alpha_{j}\beta_{j}\alpha^{-1}_{j}$, where
$\alpha_{j}$’s are arbitrary paths starting at $x$ and each $\beta_{j}$ is a
loop inside one of the open sets $U_{j}\in{\mathcal{U}}$.

The next topology is lasso topology which has been introduced and studied in
[6].

######  Definition 1.2.

For any pointed topological space $(X,x_{0})$ the lasso topology on the set
$\widetilde{X}_{H}$ is defined by the collection of all the following sets as
a basis

$B_{H}([\alpha]_{H},\mathcal{U},U)=\\{[\beta]_{H}\in{\widetilde{X}_{H}}\ |\
\beta\simeq\alpha\ast\gamma\ast\delta\ for\ some\
[\gamma]\in{\pi(\mathcal{U},\alpha(1))}$ $\ \ \ \ \ \ \ and\ for\ some\
\delta:I\rightarrow U,\ \delta(0)=\alpha(1)\\},$

where $[\alpha]_{H}\in{\widetilde{X}_{H}}$, $\mathcal{U}$ is an open cover of
$X$ and $U\in{\mathcal{U}}$ is an open neighborhood of $\alpha(1)$. In the
case $H=1$, we denote the basis elements of $\widetilde{X}_{e}$ by
$B([\alpha],\mathcal{U},U)$.

We denote the space $\widetilde{X}_{H}$ equipped with whisker and lasso
topologies by $\widetilde{X}^{wh}_{H}$ and $\widetilde{X}^{l}_{H}$,
respectively. Moreover, we denote $\widetilde{X}^{top}_{e}$,
$\widetilde{X}^{wh}_{e}$ and $\widetilde{X}^{l}_{e}$ instead of
$\widetilde{X}^{top}_{H}$, $\widetilde{X}^{wh}_{H}$ and
$\widetilde{X}^{l}_{H}$, respectively, when $H$ is the trivial subgroup. Note
that $\pi_{1}^{qtop}(X,x_{0})$ and $\pi_{1}^{wh}(X,x_{0})$ and
$\pi_{1}^{l}(X,x_{0})$ can be considered as subspaces of
$\widetilde{X}^{top}_{e}$, $\widetilde{X}^{wh}_{e}$ and
$\widetilde{X}^{l}_{e}$, respectively. The relation between these three
different topologies are as follows, when $X$ is a connected, locally path
connected space (see [18]).

“$\widetilde{X}^{wh}_{e}$ is finer than $\widetilde{X}^{top}_{e}$” and “
$\widetilde{X}^{top}_{e}$ is finer than $\widetilde{X}^{l}_{e}$”

Note that similar statements to the above hold for $\widetilde{X}_{H}$ when
$H$ is a nontrivial subgroup (see [4]).

Small loop transfer (SLT for short) spaces were introduced for the first time
by Brodskiy et al. [7, Definition 4.7]. The main motivation of the definition
of SLT spaces is to determine the condition for coincidence of the compact-
open topology and the whisker topology on $\widetilde{X}_{e}$. Indeed,
Brodskiy et al. [7, Theorems 4.11, 4.12] proved that a locally path connected
space $X$ is an SLT space if and only if for every $x\in{X}$,
$\widetilde{X}^{top}_{e}=\widetilde{X}^{wh}_{e}$. Also, they defined a strong
version of this notion, strong SLT space [7, Definition 4.18], and showed that
a path connected space $X$ is a strong SLT space if and only if for every
$x\in{X}$, $\widetilde{X}^{l}_{e}=\widetilde{X}^{wh}_{e}$ [7, Proposition
4.19]. Moreover, Pashaei et al. [13] introduced and studied SLT spaces with
respect to a subgroups $H$ of $\pi_{1}(X,x_{0})$ ($H$-SLT for short) at point
$x_{0}$, and using this notion, presented a condition for the coincidence of
the whisker and the compact-open topology on $\widetilde{X}_{H}$. In this
paper by introducing a relative version of strong small loop transfer spaces
with respect to a subgroup $H$ of $\pi_{1}(X,x_{0})$ at a point $x_{0}$
(strong $H$-SLT at $x_{0}$), we are going to determine when the whisker and
the lasso topologies on $\widetilde{X}_{H}$ are identical. Also, we study the
relationship between covering and semicovering spaces of strong $H$-SLT spaces
at a point in $X$.

######  Definition 1.3.

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$. A topological space $X$ is called
strong $H$-small loop transfer (strong $H$-SLT for short) space at $x_{0}$ if
for every $x\in X$ and for every open neighborhood $U$ of $X$ containing
$x_{0}$ there is an open neighborhood $V$ containing $x$ such that for every
loop $\beta:I\rightarrow V$ based at $x$ and for every path
$\alpha:I\rightarrow X$ from $x_{0}$ to $x$ there is a loop
$\lambda:I\rightarrow U$ based at $x_{0}$ such that,
$[\alpha\ast\beta\ast\alpha^{-1}]_{H}=[\lambda]_{H}$. Also, $X$ is called a
strong $H$-SLT space if for every $x\in{X}$ and for every path $\delta$ from
$x_{0}$ to $x$, $X$ is a strong $[\delta^{-1}H\delta]$-SLT space at $x$. Note
that if $H$ is the trivial subgroup, then a strong $H$-SLT space is a strong
SLT space.

It is well known that covering spaces of a path connected, locally path
connected and semilocally simply connected space $X$ are classified by
subgroups of the fundamental group $\pi_{1}(X,x_{0})$. When $X$ has more
complicated local structure, there need not be a simply connected cover
corresponding to the trivial subgroup. Many people have attempted to extend
the covering-theoretic approach to more general spaces. A common approach is
to designate those properties of a covering map which are assumed important.
One of them is semicoverings [2] which are defined to be local homeomorphisms
with continuous lifting of paths and homotopies which are related to
topological group structures on fundamental groups [3, 10]. The other one is
generalized universal coverings which were introduced by Fischer and Zastrow
[9] and provide combinatorial information about fundamental groups of spaces
which are not semilocally simply connected such as the Hawaiian earring, the
Menger curve, and the Sierpinski carpet.

The following theorem determines the existence of coverings for locally path
connected spaces via Spanier groups [14, Theorem 2.5.13].

######  Theorem 1.4.

Let $X$ be connected, locally path connected and $H\leq\pi_{1}(X,x_{0})$. Then
there exists a covering map $p:\widetilde{X}\rightarrow X$ with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H$ if and only if there is an
open cover $\mathcal{U}$ of $X$ such that $\pi(\mathcal{U},x_{0})\leq H$.

Brazas [3, Theorem 4.8] and Torabi et al. [15, Theorem 2.1] have addressed the
existence of covering spaces of locally path connected via algebraic and
topological structures of fundamental groups. For instance, in [15, Theorem
2.1] it was shown that any open normal subgroup in $\pi_{1}^{qtop}(X,x_{0})$
contains a Spanier group. In Section 2, we show the existence of covering
spaces of a strong $H$-SLT space at point $x_{0}$ provided that $H$ is open in
$\pi_{1}^{qtop}(X,x_{0})$. In fact, we prove that if $K$ is an open subgroup
in $\pi_{1}^{qtop}(X,x_{0})$ containing $H$, then there is an open cover
$\mathcal{U}$ of $X$ such that $\pi(\mathcal{U},x_{0})$ is contained in $K$
when $X$ is a strong $H$-SLT space at $x_{0}$ (see Proposition 2.1).

It is obvious that every covering map $p:\widetilde{X}\rightarrow X$ is a
semicovering map but not vice versa [10]. Brazas showed that if $X$ is a
connected, locally path connected and semilocally simply connected space, then
these two concepts are the same (see [2, Corollary 7.2]). Moreover, Torabi et
al. [15, Theorem 4.4] proved this fact for connected, locally path connected
and semilocally small generated spaces. In Corollary 2.3, we show that if
$p:(\widetilde{X},\tilde{x}_{0})\rightarrow(X,x_{0})$ is a semicovering map
with $p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H\leq\pi_{1}(X,x_{0})$,
then $p$ is a covering map when $X$ is a connected, locally path connected and
strong $H$-SLT space at $x_{0}$. Consequently, we can show that every
semicovering map is a covering map in strong SLT spaces at $x_{0}$. Recall
that Brazas introduced the notion of $\textbf{lpc}_{0}$-covering maps in terms
of unique lifting property [4, Definition 5.3]. Note that
$\textbf{lpc}_{0}$-covering maps were inspired by the concept of generalized
universal covering maps which introduced by Fischer and Zastrow [9]. By the
definition, it can be easily seen that every covering map is an
$\textbf{lpc}_{0}$-covering map. Since the fibers of a covering map are
discrete, Example 4.15 in [9] implies that an $\textbf{lpc}_{0}$-covering map
is not necessarily a covering map. In Proposition 2.5, we prove that if $X$ is
a strong $H$-SLT space at $x_{0}$, then an $\textbf{lpc}_{0}$-covering map
$p:(\widetilde{X},\tilde{x}_{0})\rightarrow(X,x_{0})$ with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H$ is a covering map when the
fiber $p^{-1}(x_{0})$ is finite. Finally, we address the relationship between
some famous subgroups of $\pi_{1}(X,x_{0})$ in strong SLT and SLT spaces at
$x_{0}$.

The aim of Section 3 is to clarify the relationship between the whisker
topology and the lasso topology on $\widetilde{X}_{H}$. We show that these two
topologies on $\widetilde{X}_{H}$ are identical if and only if the space $X$
is a strong $H$-SLT space when $H$ is a normal subgroup (see Theorem 3.2).
Moreover, we show that if $X$ is strong $H$-SLT at $x_{0}$, then all open
subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$ containing
normal subgroup $H$ are the same. In the case that $H$ is not normal, we show
that open subgroups of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$
containing $H$ are the same (see Proposition 3.6). However, we prove that if
$X$ is a strong $H$-SLT space at $x_{0}$, then closed normal subgroups of
$\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$ containing $H$ are the
same. In Corollary 3.11, we show that a semicovering map can transfer the
property of being strong SLT from its codomain to its domain. Finally, in
order to justify the definition of strong $H$-SLT spaces, we give an example
of an strong $H$-SLT space which is not strong SLT and consequently, it is not
semilocally simply connected (see Example 3.13). Also, we give an example to
show that some results of the paper do not necessarily hold, for instance
Proposition 3.6, for $H$-SLT spaces at $x_{0}$ (see Example 3.14).

##  2 Relationship Between Strong SLT Spaces and Covering Maps

Since existence of covering maps have a significant relation with Spanier
groups (see Theorem 1.4), it is interesting to find conditions under which for
a subgroup $H$ of $\pi_{1}(X,x_{0})$ there is an open cover $\mathcal{U}$ of
$X$ such that $\pi(\mathcal{U},x_{0})\leq H$. Recall that Torabi et al. in
[15, Theorem 2.1] proved that if $H$ is an open normal subgroup of
$\pi_{1}^{qtop}(X,x_{0})$, then there is a Spanier group which is contained in
$H$. In the following proposition, we show that if $X$ is a strong $H$-SLT
spaces at $x_{0}$, then any open subgroup of $\pi_{1}^{qtop}(X,x_{0})$
containing $H$ contains a Spanier group.

######  Proposition 2.1.

Let $H\leq\pi_{1}(X,x_{0})$ and $X$ be a connected, locally path connected and
strong $H$-SLT space at $x_{0}$. If $K$ is an open subgroup of
$\pi_{1}^{qtop}(X,x_{0})$ containing $H$, then there is an open cover
$\mathcal{U}$ of $X$ such that $\pi(\mathcal{U},x)\leq K$, i.e., there exists
a covering map $p:\widetilde{X}\rightarrow X$ with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=K$.

###### Proof.

Let $K$ be an open subgroup of $\pi_{1}^{qtop}(X,x_{0})$ containing $H$. By
the definition of the quotient topology $\pi_{1}^{qtop}(X,x_{0})$, there is an
open basis neighborhood $W=\bigcap_{i=1}^{n}\langle I_{i},U_{i}\rangle$ of the
constant path $c_{x_{0}}$ in $\Omega(X,x_{0})=\\{\alpha\ |\
\alpha(0)=\alpha(1)=x_{0}\\}$ such that $W\subseteq\pi^{-1}(K)$, where
$\pi:\Omega(X,x_{0})\rightarrow\pi_{1}(X,x_{0})$ is the quotient map defined
by $\pi(\alpha)=[\alpha]$. Put $U=\bigcap_{i=1}^{n}U_{i}$. By the structure of
$W$, we have $x_{0}\in{U}$ and so $U\neq\emptyset$. Since $X$ is a connected,
locally path connected and strong $H$-SLT space at $x_{0}$, for every
$x\in{X}$ there is a path connected open neighborhood $V$ containing $x$ such
that for every loop $\beta:I\rightarrow V$ based at $x$ and for every path
$\alpha:I\rightarrow X$ from $x_{0}$ to $x$, there is a loop
$\lambda:I\rightarrow U$ based at $x_{0}$ such that
$[\alpha\ast\beta\ast\alpha^{-1}]_{H}=[\lambda]_{H}$. Assume $\mathcal{U}$ is
the open cover of $X$ consists of all neighborhoods $V$’s. We show that the
Spanier group $\pi(\mathcal{U},x)$ with respect to the open cover
$\mathcal{U}$ is contained in $K$. It is enough to show that this relation
holds for any generator of $\pi(\mathcal{U},x)$. Let
$[\alpha\ast\beta\ast\alpha^{-1}]$ be an arbitrary generator of
$\pi(\mathcal{U},x)$. Since all elements of $\mathcal{U}$ are path connected,
it is not hard to see that for any path $\alpha$ from $x_{0}$ to any $y\in{V}$
and for every loop $\beta$ inside $V$ based at $y$, there is a loop $\lambda$
inside $U$ such that $[\alpha\ast\beta\ast\alpha^{-1}]_{H}=[\lambda]_{H}$,
that is, $[\alpha\ast\beta\ast\alpha^{-1}\ast\lambda^{-1}]\in{H}$ or
equivalently, $[\alpha\ast\beta\ast\alpha^{-1}]\in H[\lambda]$. Since
$Im(\lambda)\subseteq U$, we have $\lambda\in{W}$, i.e.,
$\pi(\lambda)=[\lambda]\in K$. On the other hand, since $H\leq K$,
$[\alpha\ast\beta\ast\alpha^{-1}]\in K$. Hence the result holds. ∎

Recall that for any subgroup $H$ of a group $G$, the core of $H$ in $G$,
denoted by $H_{G}$, is defined to the join of all normal subgroups of $G$ that
are contained in $H$. Note that $H_{G}=\bigcap_{g\in{G}}g^{-1}Hg$ is the
largest normal subgroup of $G$ which contained in $H$. By the structure of
quasitopological group $\pi_{1}^{qtop}(X,x_{0})$, if $H_{\pi_{1}(X,x_{0})}$ is
open in $\pi_{1}^{qtop}(X,x_{0})$ then so is $H$, but the converse is not true
in general. Note that if the openness of $H$ in $\pi_{1}^{qtop}(X,x_{0})$
implies the openness of $H_{\pi_{1}(X,x_{0})}$, then Theorem 3.7 of [15]
implies that every semicovering map is a covering map. But there is a
semicovering map which is not a covering map (see [2, Example 3.8]). The
following corollary shows that the converse holds in relative version of
strong SLT spaces at one point.

######  Corollary 2.2.

Let $X$ be a connected, locally path connected and strong H-SLT space at
$x_{0}$. Then $H_{\pi_{1}(X,x_{0})}$ is open in $\pi_{1}^{qtop}(X,x_{0})$ if
and only if $H$ is an open subgroup in $\pi_{1}^{qtop}(X,x_{0})$.

###### Proof.

It is easy to see that if $H_{\pi_{1}(X,x_{0})}$ is open in
$\pi_{1}^{qtop}(X,x_{0})$, then so is $H$.

To prove the other direction, by Proposition 2.1, there is an open cover
$\mathcal{U}$ of $X$ such that $\pi(\mathcal{U},x)\leq H$. On the other hand,
since $H_{\pi_{1}(X,x_{0})}$ is the largest normal subgroup of
$\pi_{1}(X,x_{0})$ contained in $H$, we have $\pi(\mathcal{U},x)\leq
H_{\pi_{1}(X,x_{0})}$. Since $\pi(\mathcal{U},x)$ is an open subgroup and
$H_{\pi_{1}(X,x_{0})}$ is a subgroup in $\pi_{1}^{qtop}(X,x_{0})$,
$H_{\pi_{1}(X,x_{0})}$ is open in $\pi_{1}^{qtop}(X,x_{0})$. ∎

######  Corollary 2.3.

Let $H\leq\pi_{1}(X,x_{0})$ and $p:\widetilde{X}\rightarrow X$ be a
semicovering map with $p_{\ast}(\widetilde{X},\tilde{x}_{0})=H$. If $X$ is a
connected, locally path connected and strong $H$-SLT space at $x_{0}$, then
$p$ is a covering map.

###### Proof.

Let $p:\widetilde{X}\rightarrow X$ be a semicovering map with
$p_{\ast}(\widetilde{X},\tilde{x}_{0})=H$. By [3, Theorem 3.5], $H$ is an open
subgroup in $\pi_{1}^{qtop}(X,x_{0})$. Hence, Corollary 2.2 implies that
$H_{\pi_{1}(X,x_{0})}$ is open in $\pi_{1}^{qtop}(X,x_{0})$. Therefore, using
[15, Theorem 3.7], $p:\widetilde{X}\rightarrow X$ is a covering map. ∎

The classification of semicovering maps were given by Brazas in [3] when $X$
is connected and locally path connected. By the definition of a semicovering
map, it can be observed that every covering map is a semicovering map but the
converse does not hold, out of semilocally simply connected spaces [3, Example
3.8]. The following corollary shows that the converse does hold in a category
of spaces wider than semilocally simply connected spaces which is a direct
consequence of Corollary 2.3.

######  Corollary 2.4.

Let $X$ be a connected, locally path connected and strong SLT space at
$x_{0}$. Then every semicovering map is a covering map.

It turns out that every covering map is an $\textbf{lpc}_{0}$-covering map but
not vice versa. For example, the Hawiian Earring admits generalized universal
covering space [9, Proposition 3.6] but does not admit simply covering space
because it is not semilocally simply connected (see [14, Corollary 2.5.14]).

The following proposition provides some conditions under which any
$\textbf{lpc}_{0}$-covering map is a covering map.

######  Proposition 2.5.

Let $p:\widetilde{X}\rightarrow X$ be an $\textbf{lpc}_{0}$-covering map with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H\leq\pi_{1}(X,x_{0})$. Let $X$
be a strong $H$-SLT space at $x_{0}$. If $|p^{-1}(x_{0})|<\infty$, then $p$ is
a covering map.

###### Proof.

Let $p:\widetilde{X}\rightarrow X$ be an $\textbf{lpc}_{0}$-covering map with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H$. In [4, Lemma 5.10] it was
shown that $p$ is equivalent to the endpoint projection map
$p_{H}:\widetilde{X}_{H}^{wh}\rightarrow X$. Moreover, it was shown that the
fiber $p^{-1}(x_{0})$ is Hausdorff and hence $(p_{H}^{-1}(x_{0}))^{wh}$ is
Hausdorff (see [13, Corollary 3.10]). On the other hand, since
$|p^{-1}(x_{0})|<\infty$, we have $|p_{H}^{-1}(x_{0})|<\infty$. Thus,
$(p_{H}^{-1}(x_{0}))^{wh}$ is discrete. It is not hard to see that
$(p_{H}^{-1}(x_{0}))^{wh}$ agrees with $\frac{\pi_{1}^{wh}(X,x_{0})}{H}$ [13,
p. 246]. Therefore, the discreteness of $(p_{H}^{-1}(x_{0}))^{wh}$ implies
that $\frac{\pi_{1}^{wh}(X,x_{0}}{H}$ is discrete, that is, $H$ is open in
$\pi_{1}^{wh}(X,x_{0})$. Moreover, since $X$ is a strong $H$-SLT at $x_{0}$,
by [13, Proposition 3.7], $H$ is open in $\pi_{1}^{qtop}(X,x_{0})$.
Consequently, by Proposition 2.1, there exists an open cover $\mathcal{U}$ of
$X$ such that $\pi(\mathcal{U},x_{0})\leq H$ which implies that
$p:\widetilde{X}\rightarrow X$ is a covering map (see Proposition 1.4). ∎

In what follows in this section we investigate the relationship between some
subgroups of $\pi_{1}(X,x_{0})$ in strong SLT and SLT spaces. Based on some
works of [1, 12, 17] there is a chain of some effective subgroups of the
fundamental group $\pi_{1}(X,x_{0})$ as follows:

$\pi_{1}^{s}(X,x_{0})\leq\pi_{1}^{sg}(X,x_{0})\leq\widetilde{\pi}_{1}^{sp}(X,x_{0})\leq\pi_{1}^{sp}(X,x_{0}),\
\ \ \ (\star)$

where $\pi_{1}^{s}(X,x_{0})$ is the subgroup of all small loops at $x_{0}$
[17], $\pi_{1}^{sg}(X,x_{0})$ is the subgroup of all small generated loops,
i.e., the subgroup generated by the set of
$\\{[\alpha\ast\beta\ast\alpha^{-1}]\ |\
[\beta]\in{\pi_{1}^{sg}(X,\alpha(1))}\ and\ \alpha\in{P(X,x_{0})}\\}$. Also,
$\pi_{1}^{sp}(X,x_{0})$ is the Spanier group of $X$, the intersection of the
Spanier subgroups relative to open covers of $X$ [8, Definition 2.3], and
$\widetilde{\pi}_{1}^{sp}(X,x_{0})$ is the path Spanier group, i.e., the
intersection of all path Spanier subgroups
$\widetilde{\pi}(\mathcal{V},x_{0})$ [15, Definition 3.1], where $\mathcal{V}$
is a path open cover of $X$.

Recall that Jamali et al. in [11, Proposition 3.2] proved that the equality of
the first inequality of the above chain holds in SLT spaces at $x_{0}$. In the
following we are going to present some conditions for the equality of the
others inequalities.

######  Theorem 2.6.

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$ which is contained in
$\pi_{1}^{s}(X,x_{0})$. If $X$ is an $H$-SLT space at $x_{0}$, then
$\pi_{1}^{s}(X,x_{0})=\widetilde{\pi}_{1}^{sp}(X,x_{0})$.

###### Proof.

By the chain ($\star$), it is sufficient to show that
$\widetilde{\pi}_{1}^{sp}(X,x_{0})\leq\pi_{1}^{s}(X,x_{0})$. Consider
$[\lambda]\in{\widetilde{\pi}_{1}^{sp}(X,x_{0})}$. We show that $\lambda$ is a
small loop. Let $U$ be an open neighborhood of $X$ containing $x_{0}$. By
assumption, since $X$ is an $H$-SLT space at $x_{0}$, there is a path Spanier
subgroup $\widetilde{\pi}(\mathcal{V},x_{0})$ such that for
$[\lambda]\in{\widetilde{\pi}_{1}^{sp}(X,x_{0})}\leq\widetilde{\pi}(\mathcal{V},x_{0})$
there is a loop $\lambda_{U}:I\rightarrow U$ based at $x_{0}$ such that
$[\lambda]_{H}=[\lambda_{U}]_{H}$, i.e.,
$[\lambda\ast\lambda_{U}^{-1}]\in{H}$. On the other hand, since
$H\leq\pi_{1}^{s}(X,x_{0})$, there is a small loop $h:I\rightarrow U$ based at
$x_{0}$ such that $[\lambda\ast\lambda_{U}^{-1}]=[h]$, that is,
$[\lambda]=[h\ast\lambda_{U}]$ which implies that
$[\lambda]\in{\pi_{1}^{s}(X,x_{0})}$. Consequently,
$\pi_{1}^{s}(X,x_{0})=\widetilde{\pi}_{1}^{sp}(X,x_{0})$. ∎

######  Corollary 2.7.

Let $X$ be an SLT space at $x_{0}$. Then
$\pi_{1}^{s}(X,x_{0})=\widetilde{\pi}_{1}^{sp}(X,x_{0})$.

######  Theorem 2.8.

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$ which is contained in
$\pi_{1}^{s}(X,x_{0})$. If $X$ is a locally path connected strong $H$-SLT
space at $x_{0}$, then $\pi_{1}^{s}(X,x_{0})={\pi}_{1}^{sp}(X,x_{0})$.

###### Proof.

The proof is similar to that of Theorem 2.6. ∎

######  Corollary 2.9.

Let $X$ be a locally path connected strong SLT space at $x_{0}$. Then
$\pi_{1}^{s}(X,x_{0})={\pi}_{1}^{sp}(X,x_{0})$.

##  3 Relationship Between Open Subsets of $\pi_{1}^{wh}$ and $\pi_{1}^{l}$

Brodskiy et al. [7, Proposition 4.19]) showed that there is a remarkable
relation between the whisker topology and the lasso topology on
$\widetilde{X}_{e}$ in strong SLT spaces. Indeed, they showed that $X$ is a
strong SLT space if only if for every $x\in{X}$,
$\widetilde{X}_{e}^{wh}=\widetilde{X}_{e}^{l}$. Similarly, it is of interest
to determine when these topologies coincide on $\widetilde{X}_{H}$. In other
words,

“What is a necessary and sufficient condition for the equality
$\widetilde{X}_{H}^{wh}=\widetilde{X}_{H}^{l}$?”

Pashaei et al. in [13] gave a necessary and sufficient condition for the
coincidence of $\widetilde{X}_{H}^{wh}$ and $\widetilde{X}_{H}^{top}$. To
answer the above question, we need the notion of strong $H$-SLT space. For a
subgroup $H$ of $\pi_{1}(X,x_{0})$ we recall that $X$ is a strong $H$-SLT
space if for every $x\in{X}$ and for every path $\delta$ from $x_{0}$ to $x$,
$X$ is a strong $[\delta^{-1}H\delta]$-SLT space at $x$.

######  Remark 3.1.

Note that if X is a strong $H$-SLT space, then $X$ is a strong
$[\delta^{-1}H\delta]$-SLT space for every path $\delta$ from $x_{0}$ to $x$.
Let $X$ be a strong $H$-SLT space and $H$ be a normal subgroup of
$\pi_{1}(X,x_{0})$. Consider the isomorphism
$\varphi_{\delta}:\pi_{1}^{qtop}(X,x_{0})\rightarrow\pi_{1}^{qtop}(X,x)$
defined by $\varphi_{\delta}([\beta])=[\delta^{-1}\ast\beta\ast\delta]$. Then
the normality of $H$ implies that $\varphi_{\delta}(H)=[\delta^{-1}H\delta]$
is a normal subgroup of $\pi_{1}^{qtop}(X,x)$. Moreover, since
$\varphi_{\delta}$ is a homeomorphism, openness of $H$ implies that
$\varphi_{\delta}(H)=[\delta^{-1}H\delta]$ is open in $\pi_{1}^{qtop}(X,x)$.

######  Theorem 3.2.

Let $X$ be a path connected space and $H$ be a normal subgroup of
$\pi_{1}(X,x_{0})$. Then $X$ is a strong $H$-SLT space if and only if
$\widetilde{X}^{l}_{[\delta^{-1}H\delta]}=\widetilde{X}^{wh}_{[\delta^{-1}H\delta]}$
for every $x\in{X}$ and for any path $\delta$ from $x_{0}$ to $x$.

###### Proof.

Let $X$ be a strong $H$-SLT space. By definitions, it is routine to check that
$\widetilde{X}^{wh}_{K}$ is finer than $\widetilde{X}^{l}_{K}$ for any
subgroup $K\leq\pi_{1}(X,y)$. By Remark 3.1, it is sufficient to show that
$\widetilde{X}^{l}_{H}$ is finer than $\widetilde{X}^{wh}_{H}$. Consider an
open basis neighborhood $B_{H}([\alpha]_{H},U)$ in $\widetilde{X}^{wh}_{H}$,
where $\alpha$ is a path from $x_{0}$ to $x$. We find an open basis
neighborhood $B_{H}([\alpha]_{H},\mathcal{U},W)$ in $\widetilde{X}^{l}_{H}$
which is contained in $B_{H}([\alpha]_{H},U)$. By the definition of strong
$H$-SLT space, any point of $X$ has an open neighborhood $V$ defined by the
strong $H$-SLT space property which is applied to the point $\alpha(1)=x$ and
$U$, that is, for every loop $\gamma$ in $V$ and for every path $\sigma$ from
$x$ to $\gamma(0)$ there is a loop $\lambda$ in $U$ based at $x$ such that
$[\sigma\ast\gamma\ast\sigma^{-1}]_{[\alpha^{-1}H\alpha]}=[\lambda]_{[\alpha^{-1}H\alpha]}$.
Let $\mathcal{U}$ be the open cover of $X$ consisting of all neighborhoods
$V$’s. Put $W=U$ and consider $[\alpha\ast
l\ast\beta]_{H}\in{B_{H}([\alpha]_{H},\mathcal{U},U)}$. Assume $l$ is equal to
a finite concatenation of loops
$l=\prod_{i=1}^{n}\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}$, where
$\alpha_{i}$’s are paths from $x$ to $\alpha_{i}(1)$ and $\gamma_{i}$’s are
loops in some $V\in{\mathcal{U}}$ based at $\alpha_{i}(1)$. Since $X$ is
strong $H$-SLT, there are loops $\lambda_{i}$ in $U$ for $1\leq i\leq n$ such
that
$[\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}]_{[\alpha^{-1}H\alpha]}=[\lambda_{i}]_{[\alpha^{-1}H\alpha]}$,
i.e.,
$[\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\lambda^{-1}_{i}]\in{[\alpha^{-1}H\alpha]}$
for $1\leq i\leq n$. We have
$\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha_{i}^{-1}\ast\alpha^{-1}\simeq
h_{i}$ rel $\dot{I}$, where $[h_{i}]\in{H}$ ($1\leq i\leq n$). It is enough to
show that
$\big{[}\alpha\ast\big{(}\prod_{i=1}^{n}\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\big{)}\ast\beta\big{]}_{H}=\big{[}\alpha\ast\lambda\ast\beta\big{]}_{H}$,
or equivalently, it is sufficient to show that
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}\in{H}$,
where $\lambda=\lambda_{1}\ast\lambda_{2}\ast...\ast\lambda_{n}$ is a loop in
$U$ based at $x$. Crearly, it can be seen that
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}=[(\alpha\ast\alpha_{1}\ast\gamma_{1}\ast\alpha_{1}^{-1}\ast\lambda_{1}^{-1}\ast\alpha^{-1})\ast\alpha\ast\lambda_{1}\ast\alpha^{-1}\ast(\alpha\ast\alpha_{2}\ast\gamma_{2}\ast\alpha_{2}^{-1}\ast\lambda_{2}^{-1}\ast\alpha^{-1})\ast\alpha\ast\lambda_{2}\ast\alpha^{-1}....\ast(\alpha\ast\alpha_{n}\ast\gamma_{n}\ast\alpha_{n}^{-1}\ast\lambda_{n}^{-1}\ast\alpha^{-1})\ast\alpha\ast\lambda_{n}\ast\alpha^{-1}\ast(\alpha\ast\lambda_{n}^{-1}\ast\alpha^{-1})\ast(\alpha\ast\lambda_{n-1}^{-1}\ast\alpha^{-1})\ast...\ast(\alpha\ast\lambda_{1}^{-1}\ast\alpha^{-1})]$.
However, by the above observations, we have
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}=[h_{1}\ast(\alpha\ast\lambda_{1}\ast\alpha^{-1})\ast
h_{2}\ast(\alpha\ast\lambda_{2}\ast\alpha^{-1})....\ast
h_{n}\ast(\alpha\ast\lambda_{n}\ast\alpha^{-1})\ast(\alpha\ast\lambda_{n}^{-1}\ast\alpha^{-1})\ast(\alpha\ast\lambda_{n-1}^{-1}\ast\alpha^{-1})\ast...\ast(\alpha\ast\lambda_{1}^{-1}\ast\alpha^{-1})]$.
By normality of $H$, it is easy but laborious to obtain $[h]\in{H}$ such that
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}=[h_{1}\ast(\alpha\ast\gamma_{1}\ast\alpha^{-1})\ast
h\ast(\alpha\ast\gamma_{1}^{-1}\ast\alpha^{-1})]$. Hence
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}=[h_{1}\ast
h^{\prime}]$ which implies that
$\big{[}\big{(}\prod_{i=1}^{n}\alpha\ast\alpha_{i}\ast\gamma_{i}\ast\alpha^{-1}_{i}\ast\alpha^{-1}\big{)}\ast\alpha\ast\lambda^{-1}\ast\alpha^{-1}\big{]}\in{H}$,
where $[\alpha\ast\gamma_{1}\ast\alpha^{-1}\ast
h\ast\alpha\ast\gamma_{1}^{-1}\ast\alpha^{-1}]=[h^{\prime}]$.

Conversely, let $x\in{X}$ and $\delta$ be a path from $x_{0}$ to $x$. It is
enough to show that $X$ is a strong $[\delta^{-1}H\delta]$-SLT space at $x$.
Let $U$ be an open neighborhood in $X$ containing $x$. Consider open basis
neighborhood
$B_{[\delta^{-1}H\delta]}\big{(}[c_{x}]_{[\delta^{-1}H\delta]},U\big{)}$ in
$\widetilde{X}^{wh}_{[\delta^{-1}H\delta]}$. By
$\widetilde{X}^{l}_{[\delta^{-1}H\delta]}=\widetilde{X}^{wh}_{[\delta^{-1}H\delta]}$,
there is an open basis neighborhood
$B_{[\delta^{-1}H\delta]}\big{(}[c_{x}]_{[\delta^{-1}H\delta]},\mathcal{U},W\big{)}$
in $\widetilde{X}^{l}_{[\delta^{-1}H\delta]}$ such that
$B_{[\delta^{-1}H\delta]}\big{(}[c_{x}]_{[\delta^{-1}H\delta]},\mathcal{U},W\big{)}\subseteq
B_{[\delta^{-1}H\delta]}\big{(}[c_{x}]_{[\delta^{-1}H\delta]},U\big{)}$. Pick
$y\in{X}$. We know that there is an open neighborhood $V$ belonging to
$\mathcal{U}$ such that $y\in{V}$. Let $\alpha$ be a path from $x$ to $y$ and
$\beta$ be a loop in $V$ based at $y$. By the above relation,
$[c_{x}\ast\alpha\ast\beta\ast\alpha^{-1}\ast
w]_{[\delta^{-1}H\delta]}\in{B_{[\delta^{-1}H\delta]}\big{(}[c_{x}]_{[\delta^{-1}H\delta]},U\big{)}}$.
Put $w=c_{x}$. Therefore, there is a loop $\lambda$ in $U$ based at $x$ such
that
$[\alpha\ast\beta\ast\alpha^{-1}]_{[\delta^{-1}H\delta]}=[\lambda]_{[\delta^{-1}H\delta]}$
which implies that $X$ is a strong $H$-SLT space. ∎

One can easily get the following corollary of Theorem 3.2.

######  Corollary 3.3.

Let $X$ be path connected and $H$ be any normal subgroup of
$\pi_{1}(X,x_{0})$. Then $X$ is a strong $H$-SLT at $x_{0}$ if and only if
$(p^{-1}_{H}(x_{0}))^{wh}=(p^{-1}_{H}(x_{0}))^{l}$ or equivalently,
$\frac{\pi_{1}^{wh}(X,x_{0})}{H}=\frac{\pi_{1}^{l}(X,x_{0})}{H}$.

Using Corollary 3.3, one of the main result of this section can be concluded
as follows.

######  Corollary 3.4.

Let $H$ be any normal subgroup of $\pi_{1}(X,x_{0})$. If $X$ is a path
connected strong $H$-SLT space at $x_{0}$, then any subset $U$ of
$\pi_{1}(X,x_{0})$ containing $H$ is open in $\pi_{1}^{wh}(X,x_{0})$ if and
only if it is open in $\pi_{1}^{l}(X,x_{0})$.

The following proposition shows that the property of being strong $H$-SLT is a
necessary condition for the openness of the subgroup $H$ in
$\pi_{1}^{l}(X,x_{0})$.

######  Proposition 3.5.

Let $H$ be an open subgroup of $\pi_{1}^{l}(X,x_{0})$. Then $X$ is strong
$H$-SLT space.

###### Proof.

Since $H$ is an open subgroup of $\pi_{1}^{l}(X,x_{0})$, there is an open
basis neighborhood $B([c_{x_{0}}],\mathcal{U},W)$ in $\pi_{1}^{l}(X,x_{0})$
which is contained in $H$. Let $\delta$ be a path from $x_{0}$ to $x$ and $U$
be an open neighborhood containing $x$. Pick $y\in{X}$. By the definition of
open cover $\mathcal{U}$, there is an open neighborhood $V$ in $\mathcal{U}$
containing $y$. Let $\alpha$ be a path from $x$ to $y$ and $\beta$ be a loop
inside $V$ based at $y$. By the definition of $B([c_{x_{0}}],\mathcal{U},W)$,
we have
$[c_{x_{0}}\ast\delta\ast\alpha\ast\beta\ast\alpha^{-1}\ast\delta^{-1}\ast
w]\in{H}$. Put $w=c_{x_{0}}$. Hence,
$[\delta\ast\alpha\ast\beta\ast\alpha^{-1}\ast\delta^{-1}]\in{H}$ or
equivalently, $[\alpha\ast\beta\ast\alpha^{-1}]\in{[\delta^{-1}H\delta]}$,
that is,
$[\alpha\ast\beta\ast\alpha^{-1}]_{[\delta^{-1}H\delta]}=[c_{x}]_{[\delta^{-1}H\delta]}$.
Therefore, $X$ is a strong $H$-SLT space. ∎

######  Proposition 3.6.

Let $H\leq\pi_{1}(X,x_{0})$ and $X$ be a locally path connected strong $H$-SLT
space at $x_{0}$. Then $H$ is open in $\pi_{1}^{wh}(X,x_{0})$ if and only if
$H$ is open in $\pi_{1}^{l}(X,x_{0})$.

###### Proof.

As mentioned in the introduction, $\pi_{1}^{wh}(X,x_{0})$ is finer than
$\pi_{1}^{l}(X,x_{0})$.

To prove the other direction, let $H$ be open in $\pi_{1}^{wh}(X,x_{0})$. It
is easy to check that all strong $H$-SLT spaces at $x_{0}$ are $H$-SLT space
at $x_{0}$. Using [13, Proposition 3.7], $H$ is open in
$\pi_{1}^{qtop}(X,x_{0})$. On the other hand, by Proposition 2.1, there is an
open cover $\mathcal{U}$ of $X$ such that $\pi(\mathcal{U},x_{0})\leq H$.
Choose an arbitrary element $[\alpha]\in{H}$. Consider open basis neighborhood
$B([\alpha],\mathcal{U},W)$ in $\pi_{1}^{l}(X,x_{0})$, where
$W\in{\mathcal{U}}$ containing $x_{0}$. Let $[\alpha\ast l\ast
w]\in{B([\alpha],\mathcal{U},W)}$. Since $[w]\in{\pi(\mathcal{U},x_{0})}$ and
$\pi(\mathcal{U},x_{0})\leq H$, we have $[l][w]=[l\ast w]\in{H}$. Hence, by
$[\alpha]\in{H}$, we can conclude that $B([\alpha],\mathcal{U},W)\subseteq H$.
Therefore, $H$ is open in $\pi_{1}^{l}(X,x_{0})$. ∎

In the following, we show that closeness of $H$ in $\pi_{1}^{l}(X,x_{0})$ has
a remarkable relation to the fibers of the endpoint projection map
$p_{H}:\widetilde{X}^{l}_{H}\rightarrow X$.

######  Proposition 3.7.

Let $X$ be path connected and $H$ be a normal subgroup of $\pi_{1}(X,x_{0})$.
Then $(p_{H}^{-1}(x_{0}))^{l}$ is Hausdorff if and only if $H$ is closed in
$\pi_{1}^{l}(X,x_{0})$.

###### Proof.

Let $(p_{H}^{-1}(x_{0}))^{l}$ be Hausdorff. It is sufficient to show that
$\pi_{1}(X,x_{0})\setminus H$ is open in $\pi_{1}^{l}(X,x_{0})$. Let
$[\alpha]\in{\pi_{1}(X,x_{0})\setminus H}$, that is, $[\alpha]\notin{H}$ or
equivalently, $[\alpha]_{H}\neq[c_{x_{0}}]_{H}$. Since
$(p_{H}^{-1}(x_{0}))^{l}$ is Hausdorff, there are open basis neighborhoods
$M=B_{H}([\alpha]_{H},\mathcal{U},U)$ and
$N=B_{H}([c_{x_{0}}]_{H},\mathcal{V},V)$ in $(p_{H}^{-1}(x_{0}))^{l}$ such
that $M\cap N=\emptyset$, where $U$ and $V$ are open neighborhoods containing
$x_{0}$. Consider open cover $\mathcal{W}=\\{U\cap V\ |\ U\in{\mathcal{U}}\ ,\
V\in{\mathcal{V}}\\}$ and open basis neighborhood
$B([\alpha],\mathcal{W},U\cap V)$ containing $[\alpha]$. We show that
$B([\alpha],\mathcal{W},U\cap V)\cap H=\emptyset$. By contrary, suppose
$[\alpha\ast l\ast w]\in{H}$, where $[l]\in{\pi(\mathcal{W},x_{0})}$ and $w$
is a loop inside $U\cap V$ based at $x_{0}$. Note that $[\alpha\ast l\ast
w]\in{H}$ is equivalent to $[\alpha\ast l\ast w]_{H}=[c_{x_{0}}]_{H}$, i.e.,
$[\alpha\ast l\ast w]_{H}\in{M\cap N}$ which is a contradiction.

Conversely, assume $[\alpha]_{H}\neq[\beta]_{H}$, that is,
$[\alpha\ast\beta^{-1}]\notin{H}$, where $[\alpha]_{H}$,
$[\beta]_{H}\in{(p_{H}^{-1}(x_{0}))^{l}}$. Note that since $H$ is a normal
subgroup of $\pi_{1}(X,x_{0})$, it is easy to see that
$[\alpha^{-1}\ast\beta]\notin{H}$. Since $H$ is a closed subgroup of
$\pi_{1}^{l}(X,x_{0})$, there is an open cover $\mathcal{U}$ of $X$ such that
$B([\alpha^{-1}\ast\beta],\mathcal{U},U)\cap H=\emptyset$, where
$U\in{\mathcal{U}}$. Consider open basis neighborhoods
$M=B_{H}([\beta]_{H},\mathcal{U},U)$ and
$N=B_{H}([\alpha]_{H},\mathcal{U},U)$. It is enough to show that $M\cap
N=\emptyset$. By contrary, suppose $[\gamma]_{H}\in{M\cap N}$. So, there are
$[l]$, $[s]\in{\pi(\mathcal{U},x_{0})}$ and loops $w$ and $v$ inside $U$ based
at $x_{0}$ such that $[\gamma]_{H}=[\beta\ast l\ast w]_{H}=[\alpha\ast s\ast
v]_{H}$, that is, $[\beta\ast l\ast w\ast v^{-1}\ast s^{-1}\ast\alpha]\in{H}$.
By the normality of $H$, we have $[\alpha^{-1}\ast\beta\ast l\ast w\ast
v^{-1}\ast s^{-1}]\in{H}$ or equivalently, $[\alpha^{-1}\ast\beta\ast l\ast
w\ast v^{-1}\ast s^{-1}\ast c_{x_{0}}]\in{H}$, where $[l\ast w\ast v^{-1}\ast
s^{-1}]\in{\pi(\mathcal{U},x_{0})}$ and the constant path $c_{x_{0}}$ can be
seen as loop inside $U$ based at $x_{0}$. This is a contradiction because
$B([\alpha^{-1}\ast\beta],\mathcal{U},U)\cap H=\emptyset$. ∎

######  Corollary 3.8.

Let $H$ be any normal subgroup of $\pi_{1}(X,x_{0})$ and $X$ be a locally path
connected strong $H$-SLT space at $x_{0}$. Then $H$ is closed in
$\pi_{1}^{wh}(X,x_{0})$ if and only if $H$ is closed in
$\pi_{1}^{l}(X,x_{0})$.

###### Proof.

Note that $\pi_{1}^{wh}(X,x_{0})$ is finer than $\pi_{1}^{l}(X,x_{0})$, in
general [18].

To prove the other direction, let $H$ be closed in $\pi_{1}^{wh}(X,x_{0})$. We
know that all strong $H$-SLT spaces at $x_{0}$ are $H$-SLT space at $x_{0}$.
Hence by [13, Corollary 2.8], $H$ is closed in $\pi_{1}^{qtop}(X,x_{0})$. On
the other hand, by [5, Theorem 11], $p_{H}:\widetilde{X}^{wh}_{H}\rightarrow
X$ has unique path lifting property, that is, $p_{H}$ is
$\textbf{lpc}_{0}$-covering map (see [4, Theorem 5.11]). Moreover, by [13,
Corollary 3.10], $(p_{H}^{-1}(x_{0}))^{wh}$ is Hausdorff. Therefore, Corollary
3.3 and Proposition 3.7 imply that $(p_{H}^{-1}(x_{0}))^{l}$ is Hausdorff and
accordingly, $H$ is closed in $\pi_{1}^{l}(X,x_{0})$. ∎

In [4, Lemma 5.10] it was shown that every $\textbf{lpc}_{0}$-covering map is
equivalent to a certain endpoint projection map. On the other hand, the unique
path lifting property of the endpoint projection map
$p_{H}:\widetilde{X}^{wh}_{H}\rightarrow X$ implies that $p_{H}$ is an
$\textbf{lpc}_{0}$-covering map [4, Theorem 5.11]. The main advantage of the
following theorem is that the map $p:\widetilde{X}\rightarrow X$ with
$p_{\ast}\pi_{1}(\widetilde{X},\tilde{x}_{0})=H\leq\pi_{1}(X,x_{0})$ is an
$\textbf{lpc}_{0}$-covering map when $p_{H}:\widetilde{X}^{l}_{H}\rightarrow
X$ has unique path lifting property.

######  Proposition 3.9.

Let $H\unlhd\pi_{1}(X,x_{0})$. If $X$ is a locally path connected and strong
$H$-SLT space at $x_{0}$, then the following statements are equivalent.

  * 1

. $p_{H}:\widetilde{X}^{wh}_{H}\rightarrow X$ has unique path lifting
property.

  * 2

. $p_{H}:\widetilde{X}^{l}_{H}\rightarrow X$ has unique path lifting property.

###### Proof.

$1.\Rightarrow 2.$ It follows from the statement “$\widetilde{X}^{wh}_{H}$ is
finer than $\widetilde{X}^{l}_{H}$”.

$2.\Rightarrow 1.$ The unique path lifting property of
$p_{H}:\widetilde{X}^{l}_{H}\rightarrow X$ is equivalent to the closeness of
$H$ in $\pi_{1}^{l}(X,x_{0})$ (see [6, Theorem 5.6]). So, by Corollary 3.8,
$H$ is closed in $\pi_{1}^{wh}(X,x_{0})$.Therefore, using [13, Corollary 2.8],
$H$ is closed in $\pi_{1}^{qtop}(X,x_{0})$ which implies that
$p_{H}:\widetilde{X}^{wh}_{H}\rightarrow X$ has unique path lifting property
[5, Theorem 11]. ∎

######  Proposition 3.10.

Let $p:\widetilde{X}\rightarrow X$ be a semicovering map and $\widetilde{H}$
be a subgroup of $\pi_{1}(\widetilde{X},\tilde{x}_{0})$, where
$p(\tilde{x}_{0})=x_{0}$. Let $H=p_{\ast}(\widetilde{H})$ where
$p_{\ast}:\pi_{1}(\widetilde{X},\tilde{x}_{0})\rightarrow\pi_{1}(X,x_{0})$ is
the induced homomorphism by $p$. If $X$ is a strong $H$-SLT space, then
$\widetilde{X}$ is a strong $\widetilde{H}$-SLT space.

###### Proof.

Assume $\tilde{\lambda}$ is an arbitrary path from $\tilde{x}_{0}$ to
$\tilde{\lambda}(1)=\tilde{x}$. It suffices to show that $X$ is a strong
$[(\tilde{\lambda})^{-1}H\tilde{\lambda}]$-SLT space at $\tilde{x}$. Let
$\widetilde{S}$ be an open subset in $\widetilde{X}$ containing $\tilde{x}$
and $\tilde{y}$ be an arbitrary point in $\widetilde{X}$. Put
$p\circ\tilde{\lambda}=\lambda$ and $p(\tilde{x})=x$, where $\lambda$ is a
path in $X$ from $x_{0}$ to $x$. Since $p:\widetilde{X}\rightarrow X$ is a
local homeomorphism, there is an open subset $\widetilde{W}$ of $\tilde{x}$
such that $p|_{\widetilde{W}}:\widetilde{W}\rightarrow W$ is a homeomorphism.
Put $\widetilde{U}=\widetilde{S}\cap\widetilde{W}$. Note that
$p|_{\widetilde{U}}:\widetilde{U}\rightarrow U$ is a homeomorphism as well,
where $p(\widetilde{U})=U$ is an open subset of $x$ in $X$. By assumption,
since $X$ is a strong $[\lambda^{-1}H\lambda]$-SLT space at $x$, for point
$p(\tilde{y})=y$ there is an open subset $V$ containing $y$ such that for
every path $\alpha$ from $x$ to $y$ and for every loop $\beta$ at $y$ in $V$
there is a loop $\delta$ based at $x$ in $U$ such that
$[\alpha\ast\beta\ast\alpha^{-1}]_{[\lambda^{-1}H\lambda]}=[\delta]_{[\lambda^{-1}H\lambda]}$.
By the local homeomorphism property of $p:\widetilde{X}\rightarrow X$, we have
an open subset $\widetilde{V}$ of $\tilde{y}$ such that
$p|_{\widetilde{V}}:\widetilde{V}\rightarrow V$ is a homeomorphism. Now let
$\tilde{\alpha}$ be a path from $\tilde{x}$ to $\tilde{y}$ and $\tilde{\beta}$
is a loop based at $\tilde{y}$ in $\widetilde{V}$. Put
$p\circ\tilde{\alpha}=\alpha$ and $p\circ\tilde{\beta}=\beta$, where $\alpha$
is a path from $x$ to $y$ and $\beta$ is a loop at $y$ in $V$. Hence there is
a loop $\delta:I\rightarrow U$ at $x$ such that
$[\alpha\ast\beta\ast\alpha^{-1}]_{[\lambda^{-1}H\lambda]}=[\delta]_{[\lambda^{-1}H\lambda]}$
or equivalently,
$[\lambda\ast\alpha\ast\beta\ast\alpha^{-1}\delta^{-1}\ast\lambda^{-1}]\in{H}$.
By the homeomorphism $p|_{\widetilde{U}}:\widetilde{U}\rightarrow U$, there is
a loop $\tilde{\delta}$ at $\tilde{x}$ in $\widetilde{U}$ such that
$p(\tilde{\delta})=\delta$. On the other hand, we know that
$p_{\ast}([\tilde{\lambda}\ast\tilde{\alpha}\ast\tilde{\beta}\ast(\tilde{\alpha})^{-1}\ast(\tilde{\delta})^{-1}\ast(\tilde{\lambda})^{-1}])=[(p\circ\tilde{\lambda})\ast(p\circ\tilde{\alpha})\ast(p\circ\tilde{\beta})\ast(p\circ(\tilde{\alpha})^{-1})\ast(p\circ(\tilde{\delta})^{-1})\ast(p\circ(\tilde{\lambda})^{-1})]=[\lambda\ast\alpha\ast\beta\ast\alpha^{-1}\delta^{-1}\ast\lambda^{-1}]\in{H}$.
Moreover, by the definition of semicovering map [2, Definition 3.1],
$p_{\ast}$ is injection. Thus, by the definition of $\widetilde{H}$, we have
$[\tilde{\lambda}\ast\tilde{\alpha}\ast\tilde{\beta}\ast(\tilde{\alpha})^{-1}\ast(\tilde{\delta})^{-1}\ast(\tilde{\lambda})^{-1}]\in{\widetilde{H}}$
or equivalently,
$[\tilde{\alpha}\ast\tilde{\beta}\ast(\tilde{\alpha})^{-1}]_{[(\tilde{\lambda})^{-1}H\tilde{\lambda}]}=[(\tilde{\delta})^{-1}]_{[(\tilde{\lambda})^{-1}H\tilde{\lambda}]}$.
Therefore, $\widetilde{X}$ is a strong $\widetilde{H}$-SLT space. ∎

######  Corollary 3.11.

Let $p:\widetilde{X}\rightarrow X$ be a semicovering map. If $X$ is a strong
SLT space, then so is $\widetilde{X}$.

It is known that any semilocally simply connected space is a strong SLT space
and any strong SLT space is a strong $H$-SLT space for every subgroup $H$ of
$\pi_{1}(X,x_{0})$. Note that any space $X$ is a strong $\pi_{1}(X,x_{0})$-SLT
space. The following theorem help us to give an example of a strong $H$-SLT
space which is not a strong SLT space and hence, it is not semilocally simply
connected, where $H\neq\pi_{1}(X,x_{0})$.

######  Theorem 3.12.

Let $X$ be a locally path connected space and $H$ be an open normal subgroup
in $\pi_{1}^{qtop}(X,x_{0})$. Then $X$ is a strong $H$-SLT space.

###### Proof.

Let $H$ be an open normal subgroup in $\pi_{1}^{qtop}(X,x_{0})$. By Theorem
2.1 of [15], There is an open cover $\mathcal{U}$ in $X$ such that
$\pi(\mathcal{U},x_{0})\leq H$. On the other hand, by [6, Proposition 4.4],
$\widetilde{X}^{wh}_{H}=\widetilde{X}^{l}_{H}$. So, using Remark 3.1, we can
conclude that for each path $\delta$ from $x_{0}$ to $x$,
$\widetilde{X}^{l}_{[\delta^{-1}H\delta]}=\widetilde{X}^{wh}_{[\delta^{-1}H\delta]}$.
Therefore, Theorem 3.2 implies that $X$ is a strong $H$-SLT space. ∎

The following example can justify introducing the relative version of strong
SLT spaces with respect to subgroups of the fundamental group.

######  Example 3.13.

Let $(S^{1},0)$ be the unit circle, $(HA,x)$ be the Harmonic Archipelago,
where $x$ is the common point of boundary circles. We consider the wedge space
$X=\frac{S^{1}\sqcup HA}{0\sim x}$. In [16, Example 4.4] it is shown that
$\pi_{1}(X,x_{0})\neq\pi_{1}^{sg}(X,x_{0})$. On the other hand, $X$ is a
semilocally small generated space [16]. Accordingly, $\pi_{1}^{sg}(X,x_{0})$,
introduced by Virk [17], is an open subgroup of $\pi_{1}^{qtop}(X,x_{0})$.
Using Theorem 3.12, we conclude that $X$ is a strong
$\pi_{1}^{sg}(X,x_{0})$-SLT space. It is not hard to show that $X$ is not a
strong SLT space. To prove that $X$ is not a strong SLT space, consider an
arbitrary path in $X$ inside $HA$ from any semilocally simply connected point
to the wedge point.

In the following example we show that some results of the paper does not
necessarily hold, for instance Proposition 3.7, for $H$-SLT spaces at a point.
Also, note that the example below shows that the concepts of relative version
of strong SLT and SLT spaces are not necessarily the same.

######  Example 3.14.

In [10], Fischer and Zastrow presented an open subgroup $H$ of
$\pi_{1}^{qtop}(HE,x_{0})$ which does not contain any open normal subgroup and
does not correspond to a covering space [3, Theorem 4.8], where $x_{0}$ is the
common point of circles. On the other hand, by Definition 1.2 and Proposition
1.4, it is not hard to observe that covering subgroups correspond to open
subgroups of $\pi_{1}^{l}(X,x_{0})$. Moreover, since $H$ is an open subgroup
of $\pi_{1}^{qtop}(HE,x_{0})$, Proposition 3.6 of [13] implies that $HE$ is an
$H$-SLT space at $x_{0}$ and hence, using [13, Proposition 3.7], $H$ is an
open subgroup of $\pi_{1}^{wh}(X,x_{0})$. One can see that $H$ is not an open
subgroup in $\pi_{1}^{l}(HE,x_{0})$ because it is not a covering subgroup.
Therefore, the property of $H$-SLT at $x_{0}$ is not strong enough to prove
some results of this paper. Also, we can conclude that $HE$ is an $H$-SLT
space which is not a strong $H$-SLT space.

## Reference

## References

  * [1] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi, S.Z. Pashaei, On topologized fundamental subgroups and generalized coverings, to appear in Bull. Iranian Math. Soc.
  * [2] J. Brazas, Semicoverings: a generalization of covering space theory, Homol. Homotopy Appl. 14 (2012) 33–63.
  * [3] J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44 (2014) 285–313.
  * [4] J. Brazas, Generalized covering space theories, Theory Appl. Categ. 30 (2015) 1132–1162.
  * [5] J. Brazas, P. Fabel, On fundamental group with the quotient topology, Homotopy Rel. Struc. 10 (2015) 71–91.
  * [6] N. Brodskiy, J. Dydak, B. Labuz, A. Mitra, Covering maps for locally path connected spaces, Fund. Math. 218 (2012) 13–46.
  * [7] N. Brodskiy, J. Dydak, B. Labuz, A. Mitra, Topological and uniform structures on universal covering spaces, arXiv:1206.0071.
  * [8] H. Fischer, D. Repovš, Ž. Virk, A. Zastrow, On semilocally simply connected spaces, Topol. Appl. 158 (2011) 397–408.
  * [9] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fund. Math. 197 (2007) 167–196.
  * [10] H. Fischer, A. Zastrow, A core-free semicovering of the Hawaiian Earring, Topol. Appl. 160 (2013) 1957–1967.
  * [11] N. Jamali, B. Mashayekhy, H. Torabi, S.Z. Pashaei and M. Abdullahi Rashid (2017), On Topologized Fundamental Groups with Small Loop Transfer Viewpoints, arXiv:1708.02606v1.
  * [12] B. Mashayekhy, A. Pakdaman and H. Torabi, Spanier spaces and covering theory of non-homotopically path Hausdorff spaces, Georgian Math. J. 20 (2013) 303–317.
  * [13] S.Z. Pashaei, B. Mashayekhy, H. Torabi and M. Abdullahi Rashid, Small loop transfer spaces with respect to subgroups of fundamental groups, Topol. Appl. 232 (2017) 242–255.
  * [14] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
  * [15] H. Torabi, A. Pakdaman, B. Mashayekhy, On the Spanier groups and covering and semicovering spaces, arXiv:1207.4394.
  * [16] H. Torabi, A. Pakdaman, B. Mashayekhy, Topological Fundamental Groups and Small Generated coverings, Math. Slovaca 65 (2015) 1153–1164.
  * [17] Ž. Virk, Small loop spaces, Topol. Appl. 157 (2010) 451–455.
  * [18] Ž. Virk, A. Zastrow, The comparison of topologies related to various concepts of generalized covering spaces, Topol. Appl. 170 (2014) 52–62.