#  When Is a Local Homeomorphism a Semicovering Map?

Majid Kowkabi<EMAIL_ADDRESS>Behrooz Mashayekhy<EMAIL_ADDRESS>Hamid Torabi<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

In this paper, by reviewing the concept of semicovering maps, we present some
conditions under which a local homeomorphism becomes a semicovering map. We
also obtain some conditions under which a local homeomorphism is a covering
map.

###### keywords:

local homeomorphism, fundamental group, covering map, semicovering map.

###### MSC:

[2010]57M10, 57M12, 57M05

††journal:

##  1 Introduction

It is well-known that every covering map is a local homeomorphism. The
converse seems an interesting question that when a local homeomorphism is a
covering map (see [5], [9], [7].) Recently, Brazas [3, Definition 3.1]
generalized the concept of covering map by the phrase “A semicovering map is a
local homeomorphism with continuous lifting of paths and homotopies”. Note
that a map $p:Y\to X$ has continuous lifting of paths if $\rho_{p}:(\rho
Y)_{y}\to(\rho X)_{p(y)}$ defined by $\rho_{p}(\alpha)=p\circ\alpha$ is a
homeomorphism for all $y\in Y,$ where $(\rho Y)_{y}=\\{\alpha:[0,1]\to
Y|\alpha(0)=y\\}$. Also, a map $p:Y\to X$ has continuous lifting of homotopies
if $\Phi_{p}:(\Phi Y)_{y}\to(\Phi X)_{p(y)}$ defined by
$\Phi_{p}(\phi)=p\circ\phi$ is a homeomorphism for all $y\in Y$, where
elements of $(\Phi Y)_{y}$ are endpoint preserving homotopies of paths
starting at $y$. It is easy to see that any covering map is a semicovering.

The quasitopological fundamental group $\pi_{1}^{qtop}(X,x)$ is the quotient
space of the loop space $\Omega(X,x)$ equipped with the compact-open topology
with respect to the function $\Omega(X,x)\to\pi_{1}(X,x)$ identifying path
components (see [2]). Biss [2, Theorem 5.5] showed that for a connected,
locally path connected space $X$, there is a one to one correspondence between
its equivalent classes of connected covering spaces and the conjugacy classes
of open subgroups of its fundamental group $\pi_{1}^{qtop}(X,x)$. There is a
misstep in the proof of the above theorem. Torabi et al. [12] pointed out the
above misstep and gave the true classification of connected covering spaces of
$X$ according to subgroups of the fundamental group $\pi_{1}^{qtop}(X,x)$ with
open core. Using this classification, it can be concluded that for a locally
path connected space $X$ a semicovering map $p:\tilde{X}\longrightarrow X$ is
a covering map if and only if the core of
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ in $\pi_{1}(X,x_{0})$ is an open
subgroup of $\pi_{1}^{qtop}(X,x_{0})$. By using this fact, we give some
conditions under which a semicovering map becomes a covering map which extend
some results of [5].

In Section 2, among reviewing the concept of local homeomorphism, path lifting
property and unique path lifting property, we mention a result on uniqueness
of lifting for local homeomorphism and a simplified definition [8, Corollary
2.1] for a semicovering map. Note that there is a misstep in the
simplification of semicovering which we give another proof to remedy this
defect. In Section 3, we intend to find some conditions under which a local
homeomorphism is a semicovering map. Among other things, we prove that if
$p:\tilde{X}\to X$ is a local homeomorphism, $\tilde{X}$ is Hausdorff and
sequential compact, then $p$ is a semicovering map. Also, a closed local
homeomorphism from a Hausdorff space is a semicovering map. Moreover, a proper
local homeomorphism from a Hausdorff space onto a Hausdorff space is a
semicovering..

Finally in Section 4, we generalize some results of [5]. In fact, by openness
of the core of $p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ in
$\pi_{1}(X,x_{0})$, for a local homeomorphism $p:\tilde{X}\longrightarrow X$,
we obtain some conditions under which a semicovering map is a covering map.
More precisely, we prove that every finite sheeted semicovering map on a
locally path connected space is a covering map. Also, a proper local
homeomorphism from a Hausdorff space onto a locally path connected Hausdorff
space is a covering map.

##  2 Notations and Preliminaries

In this paper, all maps $f:X\to Y$ between topological spaces $X$ and $Y$ are
continuous functions. We recall that a continuous map
$p:\tilde{X}\longrightarrow X$, is called a local homeomorphism if for every
point $\tilde{x}\in\tilde{X}$ there exists an open neighbourhood $\tilde{W}$
of $\tilde{x}$ such that $p(\tilde{W})\subset X$ is open and the restriction
map $p|_{\tilde{W}}:\tilde{W}\longrightarrow p(\tilde{W})$ is a homeomorphism.
In this paper, we denote a local homeomorphism $p:\tilde{X}\longrightarrow X$
by $(\tilde{X},p)$ and assume that $\tilde{X}$ is path connected and $p$ is
surjective.

######  Definition 2.1.

Assume that $X$ and $\tilde{X}$ are topological spaces and
$p:\tilde{X}\longrightarrow X$ is a continuous map. Let
$f:(Y,y_{0})\to(X,x_{0})$ be a continuous map and $\tilde{x}_{0}\in
p^{-1}(x_{0})$. If there exists a continuous map
$\tilde{f}:(Y,y_{0})\to(\tilde{X},\tilde{x}_{0})$ such that
$p\circ\tilde{f}=f$, then $\tilde{f}$ is called a lifting of $f$.

The map $p$ has path lifting property if for every path $f$ in $X$, there
exists a lifting $\tilde{f}:(I,0)\to(\tilde{X},\tilde{x}_{0})$ of $f$. Also,
the map $p$ has unique path lifting property if for every path $f$ in $X$,
there is at most one lifting $\tilde{f}:(I,0)\to(\tilde{X},\tilde{x}_{0})$ of
$f$ (see [10].)

The following lemma is stated in [6, Lemma 5.5] for $Y=I$. One can state it
for an arbitrary map $f:X\to Y$ for a connected space $Y$.

######  Lemma 2.2.

Let $(\tilde{X},p)$ be a local homeomorphism on $X$, $Y$ be a connected space,
$\tilde{X}$ be Hausdorff and $f:(Y,y_{0})\to(X,x_{0})$ be a continuous map.
Given $\tilde{x}\in p^{-1}(x_{0})$ there is at most one lifting
$\tilde{f}:(Y,y_{0})\to(\tilde{X},\tilde{x}_{0})$ of $f.$

The following theorem can be concluded from [8, Definition 7, Lemma 2.1,
Proposition 2.2].

######  Theorem 2.3.

([8, Corollary 2.1]). A map $p:\tilde{X}\longrightarrow X$ is a semicovering
map if and only if it is a local homeomorphism with unique path lifting and
path lifting properties.

Note that during reviewing process of the previous version of this paper, the
referee pointed out that there exists a misstep in the proof of [8, Lemma
2.1]. More precisely, in the proof of [8, Lemma 2.1], it is not guaranteed
that $h_{t}(K^{j}_{n})\cap U)\neq\phi$, i.e., $h_{t}|_{K^{j}_{n}}$ might be a
lift of $\gamma$ that is different from $(p|_{U})^{-1}\circ\gamma$. After some
attempts to find a proof for the misstep, we found out that the method in the
proof of [8, Lemma 2.1] dose not work. Now, we give a proof for [8, Lemma 2.1]
with a different method as follows.

######  Lemma 2.4.

(Local Homeomorphism Homotopy Theorem). Let $p:\tilde{X}\longrightarrow X$ be
a local homeomorphism with unique path lifting and path lifting properties.
Consider the diagram of continuous maps

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{\tilde{f}}$$\textstyle{(\tilde{X},\tilde{x}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{I\times
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\tilde{F}}$$\textstyle{(X,x_{0}),}$

where $j(t)=(t,0)$ for all $t\in I$. Then there exists a unique continuous map
$\tilde{F}:I\times I\to\tilde{X}$ making the diagram commute.

###### Proof.

Put $\tilde{F}(t,0)=\tilde{f}(t)$, for all $t\in I$, and $W_{\tilde{f}(t)}$ an
open neighborhood of $\tilde{f}(t)$ in $\tilde{X}$ such that
$p|_{W_{\tilde{f}(t)}}:W_{\tilde{f}(t)}\to p(W_{\tilde{f}(t)})$ is a
homeomorphism. Then $\\{\tilde{f}^{-1}(W_{\tilde{f}(t)})|t\in I\\}$ is an open
cover for $I$. Since $I$ is compact, there exists $n\in\mathbb{N}$ such that
for every $0\leq i\leq n-1$ the interval $[\frac{i}{n},\frac{i+1}{n}]$ is
contained in $\tilde{f}^{-1}(W_{\tilde{f}(t_{i})})$ for some $t_{i}\in I$. For
every $0\leq i\leq n-1$, $F^{-1}(p(W_{{\tilde{f}}(t_{i})}))$ is open in
$I\times I$ which contains $(\frac{i}{n},0)$. Hence there exists $s_{i}\in I$
such that $[\frac{i}{n},\frac{i+1}{n}]\times[0,s_{i}]$ is contained in
$F^{-1}(p(W_{{\tilde{f}}(t_{i})}))$ and so
$F([\frac{i}{n},\frac{i+1}{n}]\times[0,s_{i}])\subseteq
p(W_{{\tilde{f}}(t_{i})})$. Since
$p|_{W_{\tilde{f}(t_{i})}}:W_{\tilde{f}(t_{i})}\to p(W_{\tilde{f}(t_{i})})$ is
a homeomorphism, we can define $\tilde{F}$ on
$k_{i}=[\frac{i}{n},\frac{i+1}{n}]\times[0,s_{i}]$ by
$p^{-1}|_{W_{\tilde{f}(t_{i})}}\circ F|_{k_{i}}$. Let $s=\min\\{s_{i}|0\leq
i\leq n-1\\}$, then by gluing lemma, we can define $\tilde{F}$ on
$I\times[0,s]$ since $\\{\frac{i+1}{n}\\}\times[0,s]\in k_{i}\cap k_{i+1}.$
Put $A=\\{r\in I|\ \mathrm{there}\ \mathrm{exists}\
\tilde{F_{r}}:I\times[0,r]\to\tilde{X}\ \mathrm{such\ that}\
F(x,y)=p\circ\tilde{F_{r}}(x,y)\mathrm{\ for\ every\ }(x,y)\in I\times[0,r]\
\mathrm{and}\ \tilde{F}(t,0)=\tilde{f}(t)\\}.$ Note that $A$ is a nonempty
since $0\in A$. We show that there exists $M\in I$ such that $M=\max{A}$. For
this, consider an increasing sequence $\\{a_{n}\\}_{n\in\mathbb{N}}$ in $A$
such that $a_{n}\to a$. We show that $a\in A$. Let $s<a$, then there exists
$n_{s}\in\mathbb{N}$ such that $s\leq a_{n_{s}}$. We define
$H:I\times[0,a]\to\tilde{X}$ by

$H(t,s)=\begin{cases}\tilde{F}_{a_{n_{s}}}(t,s)&s<a\\\
\lambda_{a}(t)&s=a,\end{cases}$

where $\lambda_{a}(t)$ is the lifting of the path $F(I\times\\{a\\})$ starting
at $\gamma(a)$ and $\gamma$ is the lifting of the path $F(0,t)$ for $t\in I$
starting at $\tilde{f}(0)$. Note that the existence of $\lambda_{a}(t)$ and
$\gamma$ are due to path lifting property of $p$. The map $H|_{I\times[0,a)}$
is well defined and continuous since if $r_{1},r_{2}\in A$ and $r_{1}<r_{2}$,
then $\tilde{F}_{r_{2}}|_{I\times[0,r_{1})}=\tilde{F}_{r_{1}}$ and $p$ is a
local homeomorphism with unique path lifting property. Therefore
$H|_{\\{0\\}\times[0,a)}$ is a lifting of $F|_{\\{0\\}\times[0,a)}$ starting
at $\tilde{f}(0)$ which implies that $H(0,t)=\gamma(t)$ for every $0\leq t<a$.
Put $B=\\{t\in
I|H|_{\\{I\times[0,a)\\}\cup\\{[0,t]\times\\{a\\}\\}}\mathrm{is\
continuous}\\}$. We show that, there exists $0<\epsilon<1$ such that
$H|_{I\times[0,a)\cup\\{[0,\epsilon)\times\\{a\\}\\}}$ is continuous. For
this, consider $U=U_{\gamma(a)}$ an open neighborhood of $\gamma(a)$ such that
$p|_{U_{\gamma(a)}}:U_{\gamma(a)}\to p(U_{\gamma(a)})$ is a homeomorphism.
There exists $0<\epsilon<1$ such that
$F([0,\epsilon]\times[a-\epsilon,a])\subseteq p(U_{\gamma(a)})$ so we have a
lifting of $F|_{[0,\epsilon]\times[a-\epsilon,a]}$ in $U_{\gamma(a)}$ by
$p^{-1}|_{U}\circ F|_{[0,\epsilon]\times[a-\epsilon,a]}$. Note that
$p^{-1}|_{U}\circ F|_{[0,\epsilon]\times[a-\epsilon,a]}(0,t)$ and $\gamma(t)$
are two liftings of $F|_{\\{0\\}\times[a-\epsilon,a]}(0,t)$ such that
$p^{-1}|_{U}\circ F(0,a)=\lambda_{a}(0)=\gamma(a)$. By unique path lifting
property we have $p^{-1}|_{U}\circ
F|_{[0,\epsilon]\times[a-\epsilon,a]}(0,t)=\gamma(t)$ for $t\in[a-\epsilon,a]$
so $p^{-1}|_{U}\circ
F|_{[0,\epsilon]\times[a-\epsilon,a]}(0,a-\frac{\epsilon}{2})=\gamma(a-\frac{\epsilon}{2})$.
Since $H(0,a-\frac{\epsilon}{2})=\gamma(a-\frac{\epsilon}{2})$, by unique path
lifting property we have $p^{-1}|_{U}\circ
F|_{[0,\epsilon]\times[a-\epsilon,a)}=H|_{[0,\epsilon]\times[a-\epsilon,a)}$.
Note that $p^{-1}|_{U}\circ F|_{[0,\epsilon]\times[a-\epsilon,a]}(t,a)$ and
$\lambda_{a}(t)$ are two liftings of $F|_{[0,\epsilon]\times\\{a\\}}(t,a)$
such that $p^{-1}|_{U}\circ F(0,a)=\gamma(a)=\lambda_{a}(0)$. By unique path
lifting property, we have $p^{-1}\circ F(t,a)=\lambda_{a}(t)=H(t,a)$ for
$t\in[0,\epsilon]$. Hence
$H|_{I\times[0,a)\cup\\{[0,\epsilon)\times\\{a\\}\\}}$ is continuous which
implies that $B$ is nonempty. We show that $B$ has a maximum element and
$\max{B}=1$. For this, consider an increasing sequence
$\\{b_{n}\\}_{n\in\mathbb{N}}$ in $B$ such that $b_{n}\to b$. We know that $H$
is continuous on $\\{I\times[0,a)\\}\cup\\{[0,b)\times\\{a\\}\\}$. By a
similar argument for the continuity of
$H|_{I\times[0,a)\cup\\{[0,\epsilon)\times\\{a\\}\\}}$, we can prove that $H$
is continuous on $\\{I\times[0,a)\\}\cup\\{[0,b]\times\\{a\\}\\}$ and $\max
B=1$. Thus $B=I$. Therefore $a\in A$, which implies that $A$ has a maximum.
Finally, by a similar idea for constructing $\tilde{F}$ on $I\times[0,s]$ we
can show that $M=1$. Hence we have a lifting for $F$ by $p$ making the above
diagram commute. Uniqueness of $\tilde{F}$ is obtained by Lemma 2.2 since
$I\times I$ is connected. ∎

Note that there exists a local homeomorphism without unique path lifting and
path lifting properties and so it is not a semicovering map .

######  Example 2.5.

Let $\tilde{X}=([0,1]\times\\{0\\})\bigcup(\\{1/2\\}\times[0,1/2))$ with
coherent topology with respect to $\\{[0,1/2]\times\\{0\\},\
(1/2,1]\times\\{0\\},\ \\{1/2\\}\times(0,1/2)\\}$ and let $X=[0,1]$. Define
$p:\tilde{X}\to X$ by

$p(s,t)=\begin{cases}s&t=0\\\ s+1/2&s=1/2\end{cases}.$

It is routine to check that $p$ is an onto local homeomorphism which dose not
have unique path lifting and path lifting properties.

##  3 When Is a Local Homeomorphism a Semicovering map?

In this section, we obtained some conditions under which a local homeomorphism
is a semicovering map. First, we intend to show that if $p:\tilde{X}\to X$ is
a local homeomorphism, $\tilde{X}$ is Hausdorff and sequential compact, then
$p$ is a semicovering map. In order to do this, we are going to study a local
homeomorphism with a path which has no lifting.

######  Lemma 3.6.

Let $p:\tilde{X}\to X$ be a local homeomorphism, $f$ be an arbitrary path in
$X$ and $\tilde{x}_{0}\in p^{-1}(f(0))$ such that there is no lifting of $f$
starting at $\tilde{x}_{0}$. If $A_{f}=\\{t\in I\ |f|_{[0,t]}\
{\mathrm{h}as}\mathrm{\ a\ lifting}\ \hat{f}_{t}\ \mathrm{on}\ [0,t]\
\mathrm{with}\ \hat{f}_{t}(0)=\tilde{x}_{0}\\}$, then $A_{f}$ is open and
connected. Moreover, there exists $\alpha\in I$ such that $A_{f}=[0,\alpha)$.

###### Proof.

Let $\beta$ be an arbitrary element of $A_{f}$. Since $p$ is a local
homeomorphism, there exists an open neighborhood $W$ at
$\hat{f}_{\beta}(\beta)$ such that $p|_{W}:W\to p(W)$ is a homeomorphism.
Since $\hat{f}_{\beta}(\beta)\in W$, there exists an $\epsilon\in I$ such that
$f[\beta,\beta+\epsilon]$ is a subset of $p(W)$. We can define a map
$\hat{f}_{\beta+\epsilon}$ as follows:

$\hat{f}_{\beta+\epsilon}(t)=\begin{cases}\hat{f}_{\beta}(t)&t\in[0,\beta]\\\
p|_{W}^{-1}(f(t))&t\in[\beta,\beta+\epsilon]\end{cases}.$

Hence $(0,\beta+\epsilon)$ is a subset of $A_{f}$ and so $A_{f}$ is open.

Suppose $t,s\in A$. Without loss of generality we can suppose that $t\geq s$.
By the definition of $A_{f}$, there exists $\hat{f}_{t}$ and so $[0,t]$ is a
subset of $A_{f}$. Also, every point between $s$ and $t$ belongs to $A_{f}$
hence $A_{f}$ is connected. Since $A_{f}$ is open connected and $0\in A_{f}$,
there exists $\alpha\in I$ such that $A_{f}=[0,\alpha)$. ∎

Now, we prove the existence and uniqueness of a concept of a defective
lifting.

######  Lemma 3.7.

let $p:\tilde{X}\to X$ be a local homeomorphism with unique path lifting
property, $f$ be an arbitrary path in $X$ and $\tilde{x}_{0}\in p^{-1}(f(0))$,
such that there is no lifting of $f$ starting at $\tilde{x}_{0}$. Then, using
the notation of the previous lemma, there exists a unique continuous map
$\tilde{f}_{\alpha}:A_{f}=[0,\alpha)\to\tilde{X}$ such that
$p\circ\tilde{f}_{\alpha}=f|_{[0,\alpha)}$.

###### Proof.

First, we defined $\tilde{f}_{\alpha}:A_{f}=[0,\alpha)\to\tilde{X}$ by
$\tilde{f}_{\alpha}(s)=\hat{f}_{s}(s)$. The map $\tilde{f}_{\alpha}$ is well
define since if $s_{1}=s_{2}$, then by unique path lifting property of $p$ we
have $\hat{f}_{s_{1}}=\hat{f}_{s_{2}}$ and so
$\hat{f}_{s_{1}}(s_{1})=\hat{f}_{s_{2}}(s_{2})$ hence
$\tilde{f}_{\alpha}(s_{1})=\tilde{f}_{\alpha}(s_{2})$. The map
$\tilde{f}_{\alpha}$ is continuous since for any element $s$ of $A_{f}$,
$\hat{f}_{\frac{\alpha+s}{2}}$ is continuous at $s$ and
$\hat{f}_{\frac{\alpha+s}{2}}=\hat{f}_{s}$ on $[0,s]$. Thus there exists
$\epsilon>0$ such that
$\tilde{f_{\alpha}}|_{(s-\epsilon,s+\epsilon)}=\hat{f}_{\frac{\alpha+s}{2}}|_{(s-\epsilon,s+\epsilon)}$.
Hence $\tilde{f}_{\alpha}$ is continuous at $s$. For uniqueness, if there
exists $\hat{f}_{\alpha}:[0,\alpha)\to\tilde{X}$ such that
$p\circ\hat{f}_{\alpha}=f|_{[0,\alpha)}$, then by unique path lifting property
of $\tilde{X}$ we must have $\tilde{f}_{\alpha}=\hat{f}_{\alpha}$. ∎

######  Definition 3.8.

By Lemmas 3.6 and 3.7, we called $\tilde{f}_{\alpha}$ the incomplete lifting
of $f$ by $p$ starting at $\tilde{x}_{0}$.

######  Theorem 3.9.

If $\tilde{X}$ is Hausdorff and sequential compact and
$p:\tilde{X}\longrightarrow X$ is a local homeomorphism, then $p$ has the path
lifting property.

###### Proof.

Let $f:I\to X$ be a path which has no lifting starting at $\tilde{x}_{0}\in
p^{-1}(f(0))$. Using the notion of Lemma 3.6, let
$\tilde{f}:A_{f}=[0,\alpha)\to\tilde{X}$ be the incomplete lifting of $f$ at
$\tilde{x}_{0}$. Suppose $\\{t_{n}\\}_{0}^{\infty}$ is a sequence in $A_{f}$
which tends to $t_{0}$ and $t_{n}\leq t_{0}$. Since $\tilde{X}$ is a
sequential compact, there exists a convergent subsequence of
${\tilde{f}(t_{n})}$, $\\{\tilde{f}(t_{n_{k}})\\}_{0}^{\infty}$ say, such that
$\tilde{f}(t_{n_{k}})$ tends to $l$. We define

$g(t)=\begin{cases}\tilde{f}(t)&0\leq t<t_{0}\\\
l=\lim_{k\rightarrow\infty}\tilde{f}(t_{n_{k}})&t=t_{0}.\end{cases}$

We have
$p(l)=p(\lim_{k\rightarrow\infty}\tilde{f}(t_{n_{k}}))=\lim_{k\rightarrow\infty}p(\tilde{f}(t_{n_{k}}))=\lim_{k\rightarrow\infty}f(t_{n_{k}})=f(t_{0})$
and so $p\circ g=f$. We show that $g$ is continuous on $[0,t_{0}]$, for this
we show that $g$ is continuous at $t_{0}$. Since $p$ is a local homeomorphism,
there exists a neighborhood $W$ at $l$ such that $p|_{W}:W\to p(W)$ is a
homeomorphism. Hence there is $a\in I$ such that $f([a,t_{0}])\subseteq p(W)$.
Let $V$ be a neighborhood at $l$ and $W^{\prime}=V\cap W$, then
$p(W^{\prime})\subseteq p(W)$ is an open set. Put
$U=f^{-1}(p(W^{\prime}))\cap(a,t_{0}]$ which is open in $[0,t_{0}]$ at
$t_{0}$. It is enough to show that $g(U)\subseteq W^{\prime}$. Since
$f(U)\subseteq p(W^{\prime})$ and $p$ is a homeomorphism on $W^{\prime}$,
$(p|_{W})^{-1}(f(U))\subseteq(p|_{W})^{-1}(p(W^{\prime}))$ and so
$(p|_{W})^{-1}\circ f=\tilde{f}$ on $[a,t_{0})$ since $p(l)=f(t_{0})$. Hence
$(p|_{W})^{-1}\circ f=g$ on $[a,t_{0}]$. Thus $g(U)\subseteq
g(f^{-1}(p(W^{\prime})))=(p|_{W})^{-1}\circ
f(f^{-1}(p(W^{\prime})))\subseteq(p|_{W})^{-1}\circ p(W^{\prime})\subseteq
W^{\prime}\subseteq V$, so $g$ is continuous. Hence $t_{0}\in A_{f}$, which is
a contradiction. ∎

######  Corollary 3.10.

If $\tilde{X}$ is Hausdorff and sequential compact and
$p:\tilde{X}\longrightarrow X$ is a local homeomorphism, then $p$ is a
semicovering map.

###### Proof.

Since $\tilde{X}$ is Hausdorff and sequential compact, by Theorem 3.9, the map
$p$ has path lifting property and by Lemma 2.2 $p$ has unique path lifting
property. Hence Theorem 2.3 implies that $p$ is a semicovering map. ∎

Chen and Wang [5, Theorem 1] showed that a closed local homeomorphism $p$,
from a Hausdorff space $\tilde{X}$ onto a connected space $X$ , is a covering
map, when there exists at least one point $x_{0}\in X$ such that
$|p^{-1}(x_{0})|=k$, for some finite number $k$. In the following theorem, we
extend this result for semicovering map without finiteness condition on any
fiber.

######  Theorem 3.11.

Let $p$ be a closed local homeomorphism from a Hausdorff space $\tilde{X}$
onto a space $X$. Then $p$ is a semicovering map.

###### Proof.

Using Theorem 2.3, it is enough to show that $p$ has unique path lifting and
path lifting properties. By Lemma 2.2, $p$ has the unique path lifting
property. To prove the path lifting property for $p$, suppose there exists a
path $f$ in $X$ such that it has no lifting starting at $\tilde{x}_{0}\in
p^{-1}(f(0))$. Let $g:A_{f}=[0,\alpha)\to\tilde{X}$ be the incomplete lifting
of $f$ at $\tilde{x}_{0}$. Suppose $\\{t_{n}\\}_{0}^{\infty}$ is a sequence
which tends to $\alpha$. Put $B=\\{t_{n}|n\in\mathbb{N}\\}$, then
$\overline{g(B)}$ is closed in $\tilde{X}$ and so $p(\overline{g(B)})$ is
closed in $X$ since $p$ is a closed map. Since $\overline{f(B)}\subseteq
p(\overline{g(B)})$, $f(\alpha)\in p(\overline{g(B)})$ and so there exists
$\beta\in\overline{g(B)}$ such that $p(\beta)=f(\alpha)$. Since $p$ is a local
homeomorphism, there exists a neighborhood $W_{\beta}$ at $\beta$ such that
$p|_{W_{\beta}}:W_{\beta}\to p(W_{\beta})$ is a homeomorphism. Since
$\beta\in\overline{g(B)}$, there exists an $n_{k}\in\mathbb{N}$ such that
$g(t_{n_{r}})\in W_{\beta}$ for every $n_{r}\geq n_{k}$. Since $f(\alpha)\in
p(W_{\beta})$, there exists $k_{1}\in\mathbb{N}$ such that
$f([t_{n_{k_{1}}},\alpha])\subseteq p(W_{\beta})$. Put
$h=((p|_{W_{\beta}})^{-1}\circ f)|_{[t_{n_{k_{1}}},\alpha]}$, then $p\circ
g=f=p\circ h$ on $[t_{n_{k_{1}}},\alpha)$. Hence $p\circ h=p\circ g$ on
$[t_{n_{k_{1}}},\alpha)$. Since $p|_{W_{\beta}}$ is a homeomorphism,
$g(t_{n_{k_{1}}})=h(t_{n_{k_{1}}})$, thus $g=h$ on $[t_{n_{k_{1}}},\alpha)$.
Therefore the map $\bar{g}:[0,\alpha]\to\tilde{X}$ defined by

$\bar{g}(t)=\begin{cases}g(t)&0\leq t<\alpha\\\
h(\alpha)=\beta&t=\alpha\end{cases}$

is continuous and $p\circ\bar{g}=f$ on $[0,\alpha]$. Hence $\alpha\in A_{f}$,
which is a contradiction. ∎

A map $f:X\to Y$ is called proper if and only if $f^{-1}(H)$ is compact for
any compact subset $H$ of $Y$.

######  Theorem 3.12.

If $p$ is a proper local homeomorphism from a Hausdorff space $\tilde{X}$ onto
a Hausdorff space $X$, then $p$ is a semicovering map.

###### Proof.

The map $p$ is open since every local homeomorphism is an open map. Also,
every open proper map is closed (see [11, Fact 2.3]). Hence using Theorem
3.11, $p$ is a semicovering map. ∎

##  4 When Is a Semicovering Map a Covering Map?

If $p:\tilde{X}\to X$ is a semicovering, then $\pi_{1}(X,x_{0})$ acts on
$Y=p^{-1}(x_{0})$ by $\alpha\tilde{x}_{0}=\tilde{\alpha}(1)$, where
$\tilde{x}_{0}\in Y$ and $\tilde{\alpha}$ is the lifting of $\alpha$ starting
at $\tilde{x}_{0}$ (see [3]). Therefore, we can conclude that the stabilizer
of $\tilde{x}_{0}$, $\pi_{1}(X,x_{0})_{\tilde{x}_{0}}$, is equal to
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ for all $\tilde{x}_{0}\in Y$ and so
$|Y|=[\pi_{1}(X,x_{0}):p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))]$. Thus, if
$x_{0},x_{1}\in X$, $Y_{0}=p^{-1}(x_{0})$, $Y_{1}=p^{-1}(x_{1})$ and
$\tilde{X}$ is a path connected space, then $|Y_{0}|=|Y_{1}|$. Hence one can
define the concept of sheet for a semicovering map similar to covering maps.

The following example shows that there exists a local homeomorphism with
finite fibers which is not a semicovering map.

######  Example 4.13.

Let $\tilde{X}=(0,2)$ and let $X=S^{1}$. Define $p:\tilde{X}\to X$ by
$p(t)=e^{2\pi it}$. It is routine to check that $p$ is an onto local
homeomorphism which their fibers are finite but $p$ is not a semicover since

$|p^{-1}((0,1))|=2\neq 1=|p^{-1}((1,0))|.$

Note that the core of a subgroup $H$ of $G$, denoted by $H_{G}$, is defined to
be the join of all the normal subgroups of $G$ that are contained in $H$. It
is easy to see that $H_{G}=\bigcap_{g\in G}g^{-1}Hg.$

######  Theorem 4.14.

Suppose $p:(\tilde{X},\tilde{x}_{0})\to(X,x_{0})$ is a semicovering map and
$X$ is locally path connected such that
$[\pi_{1}(X,x_{0}):p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))]$ is finite, then
$p$ is a covering map.

###### Proof.

Since $p$ is a semicovering map, $p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ is
an open subgroup of $\pi_{1}^{qtop}(X,x_{0})$ (see [4, Corollary 3.4].) Put
$[\pi_{1}(X,x_{0}):p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))]=m$. It is well-
known that $p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ is a subgroup of the
normalizer of $p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ in $\pi_{1}(X,x_{0})$,
$N_{\pi_{1}(X,x_{0})}(p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0})))$ and so
$[\pi_{1}(\tilde{X},\tilde{x}_{0}):N_{\pi_{1}(X,x_{0})}(p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0})))]$
is finite. Therefore the core of $p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ in
$\pi_{1}(X,x_{0})$,
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))_{\pi_{1}(X,x_{0})}$ is the
intersection of a finitely many conjugations of
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$. Since any conjugation of
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ is open,
$p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))_{\pi_{1}(X,x_{0})}$ is also open in
$\pi_{1}^{qtop}(X,x_{0})$. By the classification of connected covering spaces
of $X$ (see [12]), there is a covering map
$q:(\tilde{Y},\tilde{y_{0}})\to(X,x_{0})$ such that
$q_{*}(\pi_{1}(\tilde{Y},\tilde{y_{0}}))=p_{*}(\pi_{1}(\tilde{X},\tilde{x}_{0}))$.
Hence $p,q$ are equivalent as semicoverings. Therefore $p$ is a covering map.
∎

The following result is an immediate consequence of the above theorem.

######  Corollary 4.15.

Every finite sheeted semicovering map on a locally path connected space $X$ is
a covering map.

Note that Brazas presented an infinite sheeted semicovering map which is not a
covering map (see [3, Example 3.8]). Moreover, Theorem 1 in [5] is an
immediate consequence of our results Theorem 3.11 and Corollary 4.15.

It is well-known that a proper local homeomorphism from a Hausdorff space to a
locally compact, Hausdorff space is covering map. Chen and Wang [5, Corollary
2] showed that a proper local homeomorphism from a Hausdorff, first countable
space onto a Hausdorff, connected space is a covering map. We extend this
result without first countability as follows.

######  Theorem 4.16.

If $p$ is a proper local homeomorphism from a Hausdorff space $\tilde{X}$ onto
a locally path connected, Hausdorff space $X$, then $p$ is a finite sheeted
covering map.

###### Proof.

The local homeomorphism $p$ is a semicovering map by Theorem 3.12. Since
singletons in $X$ are compact and $p$ is proper, every fiber of $p$ is compact
in $\tilde{X}$. Since fibers of a local homeomorphism are discrete, fibers of
$p$ are compact and so they are finite. Therefore $p$ is a finite sheeted
semicovering map. Hence by Corollary 4.15, $p$ is a covering map. ∎

######  Corollary 4.17.

If $\tilde{X}$ is Hausdorff and sequential compact, $X$ is locally path
connected and $p:\tilde{X}\longrightarrow X$ is a local homeomorphism with a
finite fiber, then $p$ is a covering map.

###### Proof.

By Corollary 3.10, $p$ is a semicovering map. Hence $p$ is a finite sheeted
semicovering map which is a covering map by Corollary 4.15. ∎

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