# The quotient spaces of topological groups with a $q$-point

Li-Hong Xie<EMAIL_ADDRESS>Hai-Hua Lin<EMAIL_ADDRESS>Piyu Li
<EMAIL_ADDRESS>School of Mathematics and Computational Science, Wuyi
University, Jiangmen, Guangdong, 529000, China  School of Mathematics and
Statistics, Xuzhou Institute of Technology, Xuzhou 221018, China

###### Abstract

In this paper, we study the uniformities on the double coset spaces in
topological groups. As an implication, the quotient spaces of topological
groups with a $q$-point are studied. It mainly shows that: (1) Suppose that
$G$ is a topological group with a $q$-point and $H$ is a closed subgroup of
$G$; then the quotient space $G/H$ is an open and quasi-perfect preimage of a
metrizable space; in particular, $G/H$ is an $M$-space. (2) Suppose that $G$
is a topological group with a strict $q$-point and $H$ is a closed subgroup of
$G$; then the quotient space $G/H$ is an open and sequentially perfect
preimage of a metrizable space. (3) Suppose that $G$ is a topological group
with a strong $q$-point and $H$ is a closed subgroup of $G$; then the quotient
space $G/H$ is an open and strongly sequentially perfect preimage of a
metrizable space.

###### keywords:

Topological groups; double coset spaces; $q$-points; strict $q$-points; strong
$q$-points

###### MSC:

[2020] 54A20; 54H11; 54B15; 54C10; 54E15

##  1 Introduction

A group $G$ endowed with a topology $\tau$ is called a paratopological group
if multiplication in $G$ is continuous as a mapping of $G\times G$ to $G$,
where $G\times G$ carries the usual product topology. A topological group is a
paratopological group with continuous inversion. All topological groups in
this paper are assumed to be Hausdorff.

A Tychnoff space $X$ is Čech-complete if $X$ is homeomorphic to a
$G_{\delta}$-set in a compact space. Feathered topological groups are very
important in topological algebra. Recall that a topological group G is
feathered if $G$ has a non-empty compact set $K$ of countable character in
$G$. All Čech-complete topological groups are feathered. It is also clear that
all metrizable topological groups are feathered. It is a deep result of M.M.
Choban that a feathered topological group is Čech-complete if and only if it
is Raǐkov complete. B.A. Pasynkov proved that a topological group $G$ is
feathered if and only if $G$ contains a compact subgroup $H$ such that the
quotient space $G/H$ is metrizable. By this result, we have that a topological
group $G$ is feathered iff $G$ is a paracompact $p$-space. Further more,
Arhangel’skii and Tkachenko proved that: suppose that $H$ is a closed subgroup
of a feathered group $G$. Then the quotient space $G/H$ is a paracompact space
[1, Theorem 4.3.23]. In fact, in view of the proof of this theorem, they have
proved the following result:

######  Theorem 1.1.

Suppose that $H$ is a closed subgroup of a feathered group $G$. Then the
quotient space $G/H$ is a perfect preimage of a metrizable space.

Recall that a continuous mapping $f:X\rightarrow Y$ is (quasi-) perfect if $f$
is closed and $f^{-1}(y)$ is (countably compact) compact in $X$ for each $y\in
Y$. According to E. Michael [8], a point $x\in X$ is called a $q$-point of a
space $X$ if there exists a sequence $\\{U_{n}:n\in\omega\\}$ of open
neighborhoods of $x$ in $X$ such that every sequence
$\\{x_{n}\\}_{n\in\omega}$ of points in $X$ such that $x_{n}\in U_{n}$ for
each $n\in\omega$ has a point of accumulation in $X$. We also call
$\\{U_{n}:n\in\omega\\}$ a $q$-sequence at $x$. A space $X$ is said to be a
$q$-space if every point of it is a $q$-point. Recently, Peng and Liu obtained
the following result:

######  Theorem 1.2.

[10, Theorem 6] A topological group $G$ is a $q$-space if and only if $G$ is
an open quasi-perfect preimage of a metrizable space.

In the same paper, they also introduced the class of strong $q$-spaces and
gave a characterization of strong $q$-spaces in topological groups in terms of
preimages of metrizable spaces under mappings satisfying some conditions (see
[10, Theorem 24]). A point $x\in X$ is called a strong $q$-point of a space
$X$ if there exists a sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods
of $x$ in $X$ such that every sequence $\\{x_{n}\\}_{n\in\omega}$ of points in
$X$ such that $x_{n}\in U_{n}$ for each $n\in\omega$ has a convergent
subsequence. We also call $\\{U_{n}:n\in\omega\\}$ a strong $q$-sequence at
$x$. A space $X$ is said to be a strong $q$-space if every point in $X$ is a
strong $q$-point [10].

In view of the definition of strong $q$-spaces, Lin, Xie and Chen [5]
introduced the concept of strict $q$-spaces as following: A point $x\in X$ is
called a strict $q$-point of a space $X$ if there exists a $q$-sequence
$\\{U_{n}:n\in\omega\\}$ at $x$ such that $\bigcap_{n\in\omega}U_{n}$ is a
sequentially compact set. We also call $\\{U_{n}:n\in\omega\\}$ a strict
$q$-sequence at $x$. A space $X$ is said to be a strict $q$-space if every
point in $X$ is a strict $q$-point. Lin, Xie and Chen established the
following results:

######  Theorem 1.3.

[5, Theorem 2.6] A topological group $G$ is a strong $q$-space if and only if
$G$ is an open and strongly sequential-perfect preimage of a metrizable space.

A space $X$ is called sequentially compact if every sequence of $X$ has a
convergent subsequence [2]. A continuous mapping $f:X\rightarrow Y$ is called
strongly sequential-perfect [5] if $f$ is closed and $f^{-1}(F)$ is a
sequentially compact set for each sequentially compact set $F\subseteq Y$. A
continuous mapping $f:X\rightarrow Y$ is called sequential-perfect [5] if $f$
is closed and $f^{-1}(y)$ is a sequentially compact set for each $y\in Y.$

######  Theorem 1.4.

[5, Theorem 2.4] A topological group $G$ is a strict $q$-space if and only if
$G$ is an open sequential-perfect preimage of a metrizable space.

Hence, in view of Theorems 1.1, it is natural to ask whether the quotient
spaces of topological groups, which are $q$-spaces, strong $q$-spaces or
strict $q$-spaces are certain preimages of metrizable spaces. These are the
main aims of our paper.

##  2 Preliminaries

We begin by introducing some notations. Let $X$ be a set and let $A$ and $B$
be subsets of $X\times X$, i.e. relations on the set $X$. The inverse relation
of $A$ will be denoted by $-A$, i.e., $-A=\\{(x,y):(y,x)\in A\\}$. The
composition of $A$ and $B$ will be denoted by $A+B$, i.e.,
$A+B=\\{(x,z):\text{ there is a }y\in X\text{ such that }(x,y)\in A\text{ and
}(y,z)\in B\\}.$ $2A$ means that $A+A$.

The diagonal of the Cartesian product $X\times X$ is the set
$\Delta_{X}=\\{(x,x):x\in X\\}$. Every set $V\subseteq X\times X$ that
contains $\Delta_{X}$ and satisfies the condition $V=-V$ is called an
entourage of diagonal. The family of all entourages of diagonal will be
denoted by $\mathcal{D}_{X}$. Let $x_{0}$ be a point in $X$ and let
$V\in\mathcal{D}_{X}$. The set $V[x_{0}]=\\{x\in X:(x_{0},x)\in V\\}$ is
called the ball with center $x_{0}$ and radius $V$.

A uniformity on a set $X$ is a subfamily $\mathcal{U}\subseteq\mathcal{D}_{X}$
which satisfies the following conditions:

  1. $(U1)$

If $V\in\mathcal{U}$ and $V\subseteq W\in\mathcal{D}_{X}$, then
$W\in\mathcal{U}$.

  2. $(U2)$

If $V_{1},V_{2}\in\mathcal{U}$, then $V_{1}\cap V_{2}\in\mathcal{U}$.

  3. $(U3)$

For every $V\in\mathcal{U}$, there exists a $W\in\mathcal{U}$ such that
$2W\subseteq V$.

A uniformity $\mathcal{U}$ on a set $X$ is called a separated uniformity if
$\mathcal{U}$ satisfies the condition: $(U4)$ $\bigcap\mathcal{U}=\Delta_{X}.$

A subfamily $\mathcal{B}\subseteq\mathcal{U}$ is called a base for the
uniformity $\mathcal{U}$ if for every $V\in\mathcal{U}$ there exists a
$W\in\mathcal{B}$ such that $W\subseteq V$.

The following result is well known.

######  Proposition 2.1.

Let $X$ be a set. For a subfamily $\mathcal{B}\subseteq\mathcal{D}_{X}$, if
$\mathcal{B}$ satisfies the following conditions

  1. $(BU1)$

if $V_{1},V_{2}\in\mathcal{B}$, there is $V\in\mathcal{B}$ such that
$V\subseteq V_{1}\cap V_{2}$;

  2. $(BU2)$

for every $V\in\mathcal{B}$, there exists a $W\in\mathcal{B}$ such that
$2W\subseteq V$;

then $\mathcal{B}$ is a base for a uniformity $\mathcal{U}$ on the set $X$. If
the family $\mathcal{B}$ satisfies the additional condition $(BU3)$
$\bigcap\mathcal{B}=\Delta_{X}$, then $\mathcal{B}$ is a base for a separated
uniformity $\mathcal{U}$ on the set $X$.

The following result is well known.

######  Theorem 2.2.

[2, Theorem 8.1.1] For every uniformity $\mathcal{U}$ on a set $X$ the family
$\mathcal{O}=\\{U\subseteq X:\text{ for every }x\in U\text{ there is a
}V\in\mathcal{U}\text{ such that }V[x]\subseteq U\\}$ is a topology on the
set.

The topology $\mathcal{O}$ is called the topology induced by the uniformity
$\mathcal{U}$. A topological space $X$ with a topology $\mathcal{T}$ is
(pseudo)metrizable if the topology $\mathcal{T}$ is induced by a
(pseudo)metric $d$ on the set $X$.

The following result is well known.

######  Theorem 2.3.

A topological space $X$ with a topology $\mathcal{T}$ is (pseudometrizable)
metrizable if and only if the topology $\mathcal{T}$ is induced by a
(uniformity) separated uniformity with a countable base.

Let $H$ and $K$ be subgroups of an abstract group $G$. Put

$K\backslash G/H=\\{KxH:x\in G\\}$

of subsets of $G$. It is clear that for any $x,y\in G$, either $KxH=KyH$ or
$KxH\cap KyH=\emptyset$. Therefore, $K\backslash G/H$ is a partition of the
group $G$. Denote by $\pi$ the canonical mapping of $G$ onto $K\backslash G/H$
defined by $\pi(x)=KxH$, for every $x\in G$. If $G$ is a topological group,
then we say double coset space is the $K\backslash G/H$ with the quotient
topology of $G$.

let $\mathscr{N}_{s}(e)$ be the family of all open symmetric neighborhoods of
the identity $e$ in a topological group $G$. Denote by $\pi$ the natural
mapping of $G$ onto the quotient space $Z=K\backslash G/H$. For every
$V\in\mathscr{N}_{s}(e)$, put

$E_{V}^{r}=\\{(\pi(x),\pi(y)):\pi(y)\in\pi(Vx)\\},$
$E_{V}^{l}=\\{(\pi(x),\pi(y)):\pi(y)\in\pi(xV)\\},$
$\mathscr{B}_{Z}^{r}=\\{E_{V}^{r}:V\in\mathscr{N}_{s}(e)\\}$

and

$\mathscr{B}_{Z}^{l}=\\{E_{V}^{l}:V\in\mathscr{N}_{s}(e)\\}.$

##  3 The uniformities on the double coset spaces in topological groups

A subgroup $H$ of a topological group $G$ is called neutral if for every open
neighborhood $U$ of the identity $e$ in $G$, there exists an open neighborhood
$V$ of $e$ such that $HV\subseteq UH$ (equivalently, $VH\subseteq HU$) [11].

A subgroup $H$ of a topological group $G$ is called strongly neutral if for
arbitrary open neighborhood $O$ of $H$ in $G$ there is an open neighborhood
$V$ of the identity such that $HV\subseteq O$ (equivalently, $VH\subseteq O$).
It is obvious that every strongly neutral subgroup is neutral

######  Proposition 3.1.

Let $K$ be a strongly neutral subgroup of topological group $G$. Then $KF$ and
$FK$ are closed in $G$ for each closed set $F$ in $G$.

###### Proof.

Take an arbitrary point $x\notin KF$. Then $G\setminus Fx^{-1}$ is an open
neighborhood of $K$, so there is an open neighborhood $V$ of the identity such
that $KV\subseteq G\setminus Fx^{-1}$. This implies that $KV\cap
Fx^{-1}=\emptyset$ which is equivalent to $Vx\cap KF=\emptyset$. This implies
that $KF$ is closed, because $Vx$ is an open neighborhood of $x$. Similarly,
one can show that $FK$ is closed. ∎

According to Proposition 3.1 one can easily obtain the following result.

######  Corollary 3.2.

Let $K$ be a strongly neutral subgroup of topological group $G$. Then natural
quotient mappings $\pi:G\rightarrow G/K$ and $q:G\rightarrow K\backslash G$
are open and closed.

######  Proposition 3.3.

Let $H$ and $K$ be subgroups of topological group $G$. Then the natural
mapping $\pi:G\rightarrow K\backslash G/H$ is open. If $K$ is strongly neutral
and $H$ is closed, then the double coset spaces $K\backslash G/H$ and
$H\backslash G/K$ are regular.

###### Proof.

If $U$ is open in $G$, so is $KUH$ in $G$. Thus $\pi^{-1}(\pi(U))=KUH$ is open
in $G$. $\pi(U)$ is open in $K\backslash G/H$, because $\pi$ is a quotient
mapping. This implies that $\pi$ is open.

Take an arbitrary point $\pi(x)\in K\backslash G/H$, where $x\in G$. Then
$\pi^{-1}(\pi(x))=KxH$ is closed in $G$ by Proposition 3.1, because $K$ is
strongly neutral and $xH$ is closed in $G$. $\pi(x)$ is closed in $G$ since
$\pi$ is a quotient mapping. This implies that $K\backslash G/H$ is a
$T_{1}$-space.

Take any point $z=\pi(x)$ in $K\backslash G/H$ for some $x\in G$ and a
neighbourhood $O$ of $z$. There exist open neighbourhoods $U$ and $V$ of the
identity $e$ in $G$ such that $\pi(Ux)\subseteq O$ and $V^{2}\subseteq U$.
Since $K$ is strongly neutral, one can choose an open symmetric neighbourhood
$W$ of $e$ such that $WK\subseteq KV$. Clearly, $\pi(Wx)$ is an open
neighbourhood of $\pi(x)$ in $K\backslash G/H$ such that $\pi(Wx)\subseteq O$,
and we claim that the closure of $\pi(Wx)$ is contained in $O$.

Indeed, suppose that $\pi(Wy)\cap\pi(Wx)\neq\emptyset$ for some $y\in G$. Then
$Wy\cap KWxH\neq\emptyset$. Thus we have:

$y\in WKWxH\subseteq KVWxH\subseteq KV^{2}xH\subseteq KUxH=\pi^{-1}\pi(Ux).$

It follows that $\pi(y)\in\pi(Ux)\subseteq O$. Since $\pi(Wy)$ is an open
neighbourhood of $\pi(y)$ in $K\backslash G/H$, we conclude that all
accumulation points of $\pi(Wx)$ lie in $O$. Thus the space $K\backslash G/H$
is regular.

Similarly, one can prove that the space $H\backslash G/K$ is regular. ∎

######  Theorem 3.4.

Let $K$ and $H$ be subgroups of a topological group $G$. If $K$ is neutral,
then the family $\mathscr{B}_{Z}^{r}$ is a base for some right uniformity
$\mathcal{U}_{Z}^{r}$ on the double coset space $Z=K\backslash G/H$, which
induces the quotient topology on $Z$. If $H$ is closed and $K$ is strongly
neutral, then the family $\mathscr{B}_{Z}^{r}$ is a base for some right
separated uniformity $\mathcal{U}_{Z}^{r}$ on the double coset space
$Z=K\backslash G/H$, which induces the quotient topology on $Z$.

###### Proof.

Let $K$ be neutral. First, to show that the family $\mathscr{B}_{Z}^{r}$ is a
base for some right uniformity $\mathcal{U}_{Z}^{r}$, it is only to prove that
the family $\mathscr{B}_{Z}^{r}$ satisfies the conditions $(BU1)$-$(BU2)$ in
Proposition 2.1. i.e.,

  1. $(BU1$)

for arbitrary $E_{V_{1}}^{r},E_{V_{2}}^{r}\in\mathscr{B}_{Z}^{r}$, there is a
$E_{V}^{r}\in\mathscr{B}_{Z}^{r}$ such that $E_{V}^{r}\subseteq
E_{V_{1}}^{r}\cap E_{V_{2}}^{r}$;

  2. $(BU2)$

for arbitrary $E_{V}^{r}\in\mathscr{B}_{Z}^{r}$, there is a
$E_{W}^{r}\in\mathscr{B}_{Z}^{r}$ such that $2E_{W}^{r}\subseteq E_{V}^{r}$.

For $(BU1)$, $E_{V_{1}\cap V_{2}}^{r}\subseteq E_{V_{1}}^{r}\cap
E_{V_{2}}^{r}$ is evident, because $E_{U}^{r}\subseteq E_{W}^{r}$ whenever
$U\subseteq W$.

For $(BU2)$, take arbitrary $E_{V}^{r}\in\mathscr{B}_{Z}^{r}$. For
$V\in\mathscr{N}_{s}(e)$, choose a $O\in\mathscr{N}_{s}(e)$ such that
$O^{2}\subseteq V$. Since $K$ is neutral, for $O$ there is a
$W\in\mathscr{N}_{s}(e)$ such that $W\subseteq O$ and $WK\subseteq KO$. We
claim that $2E_{W}^{r}\subseteq E_{V}^{r}$.

Indeed, take arbitrary $(\pi(x),\pi(y))\in 2E_{W}^{r}$. Then there is $z\in G$
such that $(\pi(x),\pi(z))\in E_{W}^{r}$ and $(\pi(z),\pi(y))\in E_{W}^{r}$,
which implies that $\pi(z)\in\pi(Wx)$ and $\pi(y)\in\pi(Wz)$, respectively.
Thus, we have $z\in KWxH$ and $y\in KWzH$. This implies that

$y\in KWzH\subseteq KWKWxHH\subseteq KKOWxHH=KOWxH\subseteq KO^{2}xH\subseteq
KVxH.$

Thus, $\pi(y)\in\pi(KVxH)=\pi(Vx)$, which implies that $(\pi(x),\pi(y))\in
E_{V}^{r}$. Therefore, we have proved $2E_{W}^{r}\subseteq E_{V}^{r}$. Thus,
we have proved that the family $\mathscr{B}_{Z}^{r}$ is a base for some right
uniformity $\mathcal{U}_{Z}^{r}$ on $Z$.

Next, we shall show that the topology $\mathcal{T}_{1}$ induced by the right
uniformity $\mathcal{U}_{Z}^{r}$ on $Z$ is compatible with the quotient
topology $\mathcal{T}_{2}$ on $Z$. On one hand, since $\pi$ is open by
Proposition 3.3, $E_{V}^{r}[\pi(x)]=\pi(Vx)$ is open in $Z$ for each $x\in G$
and each $E_{V}^{r}\in\mathscr{B}_{Z}^{r}$. This implies that
$\mathcal{T}_{1}\subseteq\mathcal{T}_{2}$, because the family
$\mathscr{B}_{Z}^{r}$ is a base for the right uniformity
$\mathcal{U}_{Z}^{r}$. On the other hand, take arbitrary non-empty open set
$O$ in $\mathcal{T}_{2}$ and arbitrary point $\pi(x)\in O$ for some $x\in G$.
Then $\pi^{-1}(O)$ is an open neighborhood of $x$ in $G$. One can find a
$V\in\mathscr{N}_{s}(e)$ such that $Vx\subseteq\pi^{-1}(O)$. Thus
$\pi(Vx)\subseteq O$ and $\pi(x)\in\pi(Vx)$. Since
$\pi(Vx)=E_{V}^{r}[\pi(x)]$, we have $O$ is in $\mathcal{T}_{1}$. This shows
that $\mathcal{T}_{2}\subseteq\mathcal{T}_{1}$.

Thus, we have proved that the topology induced by the right uniformity
$\mathcal{U}_{Z}^{r}$ on $Z$ is compatible with the quotient topology on $Z$.

If $K$ is strongly neutral and $H$ is closed, then it is only to show that
$\mathscr{B}_{Z}^{r}$ satisfies the conditions $(BU3)$ in Proposition 2.1.
i.e., $(BU3)$ $\Delta_{Z}=\bigcap\mathscr{B}_{Z}^{r}$.

For $(BU3)$, take arbitrary $\pi(x)\neq\pi(y)$ for some $x,y\in G$. Then
$x\notin KyH$. Since $H$ is closed and $K$ is strongly neutral, $KyH$ is
closed by Proposition 3.1. Furthermore, there is a $U\in\mathscr{N}_{s}(e)$
such that $Ux\cap KyH=\emptyset$, which implies $x\notin UKyH$. For $U$, one
can find a $V\in\mathscr{N}_{s}(e)$ such that $KV\subseteq UK$, because $K$ is
strongly neutral and $UK$ is a neighborhood of $K$. We claim that
$(\pi(y),\pi(x))\notin E_{V}^{r}$, which implies that
$\Delta_{Z}=\bigcap\mathscr{B}_{Z}^{r}$. Indeed, if $(\pi(y),\pi(x))\in
E_{V}^{r}$, then $\pi(x)\in\pi(Vy)$, which implies that

$x\in KVyH\subseteq UKyH.$

This is a contradiction with $x\notin UKyH.$ ∎

######  Remark 1.

From Theorem 3.4 and its proof it follows that:

  1. (1)

If $K=\\{e\\}$, then the family $\mathscr{B}_{G/H}^{r}$ is a base for some
right uniformity $\mathcal{U}_{G/H}^{r}$ on the coset space $G/H$, which
induces the quotient topology on $G/H$. In addition, if $H$ is closed, then
the right uniformity $\mathcal{U}_{G/H}^{r}$ is separated.

  2. (2)

If $H=\\{e\\}$, then the family $\mathscr{B}_{K\backslash G}^{r}$ is a base
for some right uniformity $\mathcal{U}_{K\backslash G}^{r}$ on the coset space
$K\backslash G$, which induces the quotient topology on $K\backslash G$. In
addition, if $K$ is closed, then the right uniformity $\mathcal{U}_{G/H}^{r}$
is separated.

Similarly, one can prove the following result.

######  Theorem 3.5.

Let $K$ and $H$ be subgroups of a topological group $G$. If $H$ is neutral,
then the family $\mathscr{B}_{Z}^{l}$ is a base for some left uniformity
$\mathcal{U}_{Z}^{l}$ on the double coset space $Z=K\backslash G/H$, which
induces the quotient topology on $Z$. If $K$ is closed and $H$ is strongly
neutral, then the family $\mathscr{B}_{Z}^{l}$ is a base for some left
separated uniformity $\mathcal{U}_{Z}^{l}$ on the double coset space
$Z=K\backslash G/H$, which induces the quotient topology on $Z$.

######  Remark 2.

From Theorem3.5, we can obtain that:

  1. (1)

If $K=\\{e\\}$, where $e$ is the identity in $G$, then the family
$\mathscr{B}_{G/H}^{l}$ is a base for some left uniformity
$\mathcal{U}_{G/H}^{l}$ on the quotient space $G/H$. In addition, if $H$ is
closed, then the left uniformity $\mathcal{U}_{G/H}^{l}$ is separated.

  2. (2)

If $H=\\{e\\}$, where $e$ is the identity in $G$, then the family
$\mathscr{B}_{K\backslash G}^{l}$ is a base for some left uniformity
$\mathcal{U}_{K\backslash G}^{l}$ on the quotient space $K\backslash G$. In
addition, if $K$ is closed, then the left uniformity $\mathcal{U}_{K\backslash
G}^{l}$ is separated.

######  Proposition 3.6.

Suppose that $K$ and $H$ are subgroups of a topological group $G$. If $K$ is
strongly neutral, then the natural mapping $q_{K}:G/H\rightarrow K\backslash
G/H$ defined by $q_{K}(\pi_{H}(x))=\pi(x)$, for each $x\in G$, is continuous,
open and closed, where $\pi_{H}:G\longrightarrow G/H$ and
$\pi:G\longrightarrow K\backslash G/H$ are the canonical mappings.

###### Proof.

Put $Z=K\backslash G/H$ and let $\pi:G\rightarrow Z$ be the canonical mapping.
Then $\pi=q_{K}\circ\pi_{H}$ and, since the mappings $\pi$ and $\pi_{H}$ are
open and continuous, so is $q_{K}$.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{H}}$$\scriptstyle{\pi}$$\textstyle{G/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{K}}$$\textstyle{K\backslash
G/H}$

It remains to verify that $q_{K}$ is a closed mapping. Let $z=\pi(x)$ be a
point of $Z$. Then $q_{K}^{-1}(z)=\pi_{H}(KxH)=\pi_{H}(Kx)$. Take an arbitrary
open neighborhood $O$ of $\pi_{H}(Kx)$ in $G/H$. Then
$Kx\subseteq\pi_{H}^{-1}(O)$. It is equivalent to
$K\subseteq\pi_{H}^{-1}(O)x^{-1}$. Since $K$ is strongly neutral and
$\pi_{H}^{-1}(O)x^{-1}$ is open, there is an open neighborhood $V$ at the
identity in $G$ such that $KV\subseteq\pi_{H}^{-1}(O)x^{-1}$. This implies
that $KVx\subseteq\pi_{H}^{-1}(O)$. In particular, $\pi_{H}(KVx)\subseteq O$.
It is obvious that $W=\pi(Vx)$ is an open neighborhood of $z$ in $Z$ which
satisfies

$q_{K}^{-1}(W)=q_{K}^{-1}(\pi(Vx))=\pi_{H}(\pi^{-1}(\pi(Vx)))=\pi_{H}(KVxH)=\pi_{H}(KVx)\subseteq
O.$

We have proved that, for every neighborhood $O$ of the fiber $q_{K}^{-1}(z)$
in $G/H$, there exists an open neighborhood $W$ of $z$ in $Z$ satisfying
$q_{K}^{-1}(W)\subseteq O$. Therefore, the mapping $q_{K}$ is closed. ∎

######  Proposition 3.7.

Suppose that $H$ and $K$ are subgroups in a topological group $G$ such that
$H$ is closed and $K$ is neutral. If the quotient space $K\backslash G$ is
first-countable, then the double coset space $K\backslash G/H$ is
pseudometrizable. If, in addition, $K$ is strongly neutral, then the double
coset space $K\backslash G/H$ is metrizable

###### Proof.

Put $Z=K\backslash G/H$. According to Theorem 3.4, $\mathscr{B}_{Z}^{r}$ is a
base for some right uniformity and right separated uniformity
$\mathcal{U}_{Z}^{r}$ on $Z$ whenever $K$ is neutral and strongly neutral,
respectively. Thus, to show that the double coset space $K\backslash G/H$ is
(pesudo)metrizable it is only to show that $\mathcal{U}_{Z}^{r}$ has a
countable base by Theorem 2.3.

Let $q:G\rightarrow K\backslash G$ and $\pi:G\rightarrow K\backslash G/H$ are
the natural quotient mappings. Then there is a continuous open mapping
$q_{K}:K\backslash G\rightarrow K\backslash G/H$ defined by
$q_{K}(q(x))=\pi(x)$ for each $x\in G$, because $\pi$ and $q$ are continuous
and open by Proposition 3.3.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\pi}$$\textstyle{K\backslash
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{K}}$$\textstyle{K\backslash
G/H}$

If $K\backslash G$ is first-countable, then let $\beta=\\{V_{n}:n\in\omega\\}$
be a family of open neighborhoods of $e$ such that $\\{q(V_{n}):n\in\omega\\}$
is a countable neighborhood base at $q(e)$, where $e$ is the identity in a
topological group $G$. Then every element of the family
$\gamma=\\{O_{n}:n\in\omega\\}$ is an open symmetric neighborhood of $e$ in a
topological group $G$, where $O_{n}=KV_{n}\cap(KV_{n})^{-1}$ for each
$V_{n}\in\beta$. We claim that the family
$\\{E_{O_{n}}^{r}:O_{n}\in\gamma\\}\subseteq\mathscr{B}_{Z}^{r}$ is a base of
$\mathcal{U}_{Z}^{r}$.

Since $\mathscr{B}_{Z}^{r}$ is a base of $\mathcal{U}_{Z}^{r}$, it is enough
to show that arbitrary element of $\mathscr{B}_{Z}^{r}$ contains an element of
$\\{E_{O_{n}}^{r}:O_{n}\in\gamma\\}$. Take arbitrary
$E_{V}^{r}\in\mathscr{B}_{Z}^{r}$. Then $q(V)$ is an open neighborhood of
$q(e)$, so there is a $V_{n}\in\beta$ such that $q(V_{n})\subseteq q(V)$. This
implies that $KV_{n}=q^{-1}(q(V_{n}))\subseteq q^{-1}(q(V))=KV$. Thus,
$O_{n}=KV_{n}\cap(KV_{n})^{-1}\subseteq KV_{n}\subseteq KV$. We claim that
$E_{O_{n}}^{r}\subseteq E_{V}^{r}$. Indeed, take arbitrary point
$(\pi(x),\pi(y))\in E_{O_{n}}^{r}$ for some $x,y\in G$. Then
$\pi(y)\in\pi(O_{n}x)$, which implies that $KyH\subseteq KO_{n}xH$. By
$O_{n}\subseteq KV$, we have that

$KyH\subseteq KO_{n}xH\subseteq KKVxH=KVxH,$

which implies that

$\pi(y)\in\pi(KyH)\subseteq\pi(KVxH)=\pi(Vx).$

Thus $(\pi(x),\pi(y))\in E_{V}^{r}$. ∎

In Proposition 3.7, when $K$ is the identity of $G$, we have the following
result.

######  Corollary 3.8.

Let $H$ be a closed subgroup of a first-countable topological group $G$
(without any separator axiom on $G$). Then the quotient space $G/H$ is
metrizable.

##  4 The coset spaces of generalized countably compact topological groups

In this section, we shall study the coset spaces of generalized countably
compact topological groups. First, recall some concepts as following:

######  Lemma 4.1.

Let $H$ be a subgroup of a topological group $G$. If $H$ is of countable
character $\gamma=\\{V_{n}:n\in\omega\\}$ in $G$ such that
$V^{2}_{n+1}\subseteq V_{n}$ for each $n\in\omega$, then $H$ is strongly
neutral. Furthermore, the quotient mapping $\pi$ of $G$ onto the quotient
space $H\backslash G$ is closed and open, and $H\backslash G$ is metrizable.

######  Lemma 4.2.

[6, Lemma 1.7.6] Let $\\{U_{n}:n\in\omega\\}$ be a decrease sequence of open
sets in $X$ such that
$\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$. Put
$C=\bigcap_{n\in\omega}U_{n}$. Then the following statements are equivalent:

  1. (1)

each sequence $\\{x_{n}:n\in\omega\\}$ with $x_{i}\in U_{i}$ for each
$i\in\omega$ has a cluster point in $X$;

  2. (2)

the family $\\{U_{n}:n\in\omega\\}$ is a base at $C$ in $X$ and $C$ is a
countably compact set in $X$.

Recall that a continuous mapping $f:X\rightarrow Y$ is quasi-perfect if $f$ is
closed such that $f^{-1}(y)$ is countably compact for each $y\in Y$. Recall
that a space is an $M$-space if it is a quasi-perfect preimage of a metrizable
space [9]. A point $x\in X$ is called a $q$-point of a space $X$ if there
exists a sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $x$ in $X$
such that any sequence $\\{x_{n}:n\in\omega\\}$ of points in $X$ such that
$x_{n}\in U_{n}$ for each $n\in\omega$ has a cluster point. A space $X$ is
said to be a $q$-space if every point of it is a $q$-point.

######  Theorem 4.3.

Suppose that $G$ is a topological group with a $q$-point and $H$ is a closed
subgroup of $G$. Then the quotient space $G/H$ is an open and quasi-perfect
preimage of a metrizable space. In particular, $G/H$ is an $M$-space.

###### Proof.

By the homogeneity of $G$, we can assume that the identity $e$ is a $q$-point.
Then there is a sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $e$
in $G$ satisfies that any sequence $\\{x_{n}:n\in\omega\\}$ of points such
that $x_{n}\in U_{n}$ for each $n\in\omega$ has a cluster point in $G$. We
define by induction a sequence $\\{V_{n}:n\in\omega\\}$ of symmetric open
neighborhoods of $e$ in $G$ satisfying the following conditions:

  1. (1)

$V_{0}\subseteq U_{0}$;

  2. (2)

$V_{n+1}^{2}\subseteq V_{n}$ and $V_{n+1}\subseteq U_{n+1}$ for each
$n\in\omega$.

Since $V_{n+1}^{2}\subseteq V_{n}$ and $V_{n}$ is a symmetric open
neighborhood of $e$ for each $n\in\omega$, we have
$\overline{V_{n+1}}\subseteq V_{n}$. One can easily show that the sequence
$\\{V_{n}:n\in\omega\\}$ satisfies the condition (1) in Lemma 4.2. Put
$K=\bigcap_{n\in\omega}V_{n}$. Then $K$ is a closed and countably compact
subgroup in $G$ such that the sequence $\\{V_{n}:n\in\omega\\}$ is a countable
neighborhood base at $K$ in $G$ by Lemma 4.2. Thus, by Lemma 4.1, $K$ is a
strongly neutral subgroup in $G$ and the quotient space $K\setminus G$ is
metrizable.

Let $\pi:G\rightarrow G/H$ and $q:G\rightarrow K\backslash G/H$ be natural
quotient mappings. Define a natural mapping $f:G/H\rightarrow K\backslash G/H$
by $f(\pi(x))=q(x)$ for each $x\in G$.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{G/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\backslash
G/H}$

Since $K$ is strongly neutral and $K\setminus G$ is metrizable, $K\backslash
G/H$ is metrizable by Proposition 3.7. By Proposition 3.6, $f$ is a
continuous, open and closed mapping from $G/H$ onto the metrizable space
$K\backslash G/H$. To show that $f$ is quasi-perfect it is enough to prove
that $f^{-1}(q(x))$ is countably compact for each $x\in G$. Indeed, since
$f^{-1}(q(x))=\pi(q^{-1}q(x))=\pi(KxH)=\pi(Kx)$, $f^{-1}(q(x))$ is countably
compact as a continuous image of the countably compact set $Kx$, because $K$
is countably compact. ∎

It is well known that every $M$-space is a $q$-space. Thus from Theorem 4.3 it
follows the following result:

######  Corollary 4.4.

[10, Theorem 6] A topological group $G$ is a $q$-space if and only if $G$ is
an open quasi-perfect preimage of a metrizable.

A continuous mapping $f:X\rightarrow Y$ is sequentially perfect if $f$ is
closed such that $f^{-1}(y)$ is sequentially compact for each $y$ in $Y$. A
point $x\in X$ is called a strict $q$-point of a space $X$ if there exists a
$q$-sequence $\\{U_{n}:n\in\omega\\}$ at $x$ such that
$\bigcap_{n\in\omega}U_{n}$ is a sequentially compact set. We also call
$\\{U_{n}:n\in\omega\\}$ a strict $q$-sequence. A space $X$ is said to be a
strict $q$-space if every point of it is a strict $q$-point. It is obvious
that any countably compact space which has no non-trivial convergent sequence
is a $q$-space but not a strict $q$-space(for example, the Stone-Čech
compactification of $\mathbb{N}$). A continuous mapping $f:X\rightarrow Y$ is
called sequentially perfect if $f$ is closed and $f^{-1}(y)$ is a sequentially
compact set for each $y\in Y.$

######  Theorem 4.5.

Suppose that $G$ is a topological group with a strict $q$-point and $H$ is a
closed subgroup of $G$. Then the quotient space $G/H$ is an open and
sequentially perfect preimage of a metrizable space.

###### Proof.

By the homogeneity of $G$, we can assume that the identity $e$ is a strict
$q$-point. Then there is a sequence $\\{U_{n}:n\in\omega\\}$ of open
neighborhoods of $e$ in $G$ satisfies that any sequence
$\\{x_{n}:n\in\omega\\}$ of points such that $x_{n}\in U_{n}$ for each
$n\in\omega$ has a cluster point in $G$ and $\bigcap_{n\in\omega}U_{n}$ is
sequentially compact. We define by induction a sequence
$\\{V_{n}:n\in\omega\\}$ of symmetric open neighborhoods of $e$ in $G$
satisfying the following conditions:

  1. (1)

$V_{0}\subseteq U_{0}$;

  2. (2)

$V_{n+1}^{2}\subseteq V_{n}$ and $V_{n+1}\subseteq U_{n+1}$ for each
$n\in\omega$.

Since $V_{n+1}^{2}\subseteq V_{n}$ and $V_{n}$ is a symmetric open
neighborhood of $e$ for each $n\in\omega$, we have
$\overline{V_{n+1}}\subseteq V_{n}$. One can easily show that the sequence
$\\{V_{n}:n\in\omega\\}$ satisfies the condition (1) in Lemma 4.2. Put
$K=\bigcap_{n\in\omega}V_{n}$. Then $K$ is a closed and sequentially compact
subgroup in $G$ such that the sequence $\\{V_{n}:n\in\omega\\}$ is a countable
neighborhood base at $K$ in $G$ by Lemma 4.2. Thus, by Lemma 4.1, $K$ is a
strongly neutral subgroup in $G$ and the quotient space $K\setminus G$ is
metrizable and the natural quotient mapping $p:G\rightarrow K\setminus G$ is
an open and closed mapping.

Let $\pi:G\rightarrow G/H$ and $q:G\rightarrow K\backslash G/H$ be natural
quotient mappings. Define a natural mapping $f:G/H\rightarrow K\backslash G/H$
by $f(\pi(x))=q(x)$ for each $x\in G$.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{G/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\backslash
G/H}$

Since $K$ is strongly neutral and $K\setminus G$ is metrizable, $K\backslash
G/H$ is metrizable by Proposition 3.7. By Proposition 3.6, $f$ is a
continuous, open and closed mapping from $G/H$ onto the metrizable space
$K\backslash G/H$. To show that $f$ is sequentially perfect it is enough to
prove that $f^{-1}(q(x))$ is sequentially compact for each $x\in G$. Indeed,
since $f^{-1}(q(x))=\pi(q^{-1}q(x))=\pi(KxH)=\pi(Kx)$, $f^{-1}(q(x))$ is
sequentially compact as a continuous image of the sequentially compact set
$Kx$. Note that $K$ is sequentially compact. ∎

######  Proposition 4.6.

Let $f:X\rightarrow Y$ be a sequentially perfect mapping. If $Y$ is first-
countable, then $X$ is a strict $q$-space.

###### Proof.

Take a point $x\in X$ and assume that $\\{U_{n}:n\in\omega\\}$ is a base at
$f(x)$ in $Y$. Then we claim that $\\{f^{-1}(U_{n}):n\in\omega\\}$ is a strict
$q$-sequence at $x$, which implies that $X$ is a strict $q$-space. Since $f$
is closed and continuous, one can easily show that the
$\\{f^{-1}(U_{n}):n\in\omega\\}$ is a base at $f^{-1}(f(x))$. Also,
$f^{-1}(f(x))=\bigcap_{n\in\omega}f^{-1}(U_{n})$ is a sequentially compact
set, because $f$ is sequentially perfect. Take any point $x_{n}\in
f^{-1}(U_{n})$ for each $n\in\omega$. If $\bigcap_{n\in\omega}f^{-1}(U_{n})$
contain a subsequence of the sequence $\\{x_{n}\\}_{n\in\omega},$ then one can
easily show that the sequence $\\{x_{n}\\}_{n\in\omega}$ has a convergent
subsequence, because $f^{-1}(f(x))=\bigcap_{n\in\omega}f^{-1}(U_{n})$ is a
sequence compact set. This implies that the sequence
$\\{x_{n}\\}_{n\in\omega}$ has an accumulation point. If
$\bigcap_{n\in\omega}f^{-1}(U_{n})$ contain no subsequence of the sequence
$\\{x_{n}\\}_{n\in\omega},$ without loss of generality, assuming that
$x_{n}\in f^{-1}(U_{n})\setminus\bigcap_{n\in\omega}f^{-1}(U_{n})$ for each
$n\in\omega$, then one can easily show that that the sequence
$\\{x_{n}\\}_{n\in\omega}$ has an accumulation point. Indeed, if not, then
$\\{x_{n}:n\in\omega\\}$ is a closed set. $\\{f(x_{n}):n\in\omega\\}$ is also
a closed set in $Y$, because $f$ is closed. Then
$Y\setminus\\{f(x_{n}):n\in\omega\\}$ is an open set containing $f(x)$.
However, there is no element of $\\{U_{n}:n\in\omega\\}$ contained in
$Y\setminus\\{f(x_{n}):n\in\omega\\}$. This is contradict with the fact that
$\\{U_{n}:n\in\omega\\}$ is a base at $f(x)$ in $Y$. Thus we have proved that
$\\{f^{-1}(U_{n}):n\in\omega\\}$ is a strict $q$-sequence. ∎

From Theorem 4.5 and Proposition 4.6 it follows that:

######  Corollary 4.7.

[5, Theorem 2.6] A topological group $G$ is a strict $q$-space if and only if
$G$ is an open and sequentially perfect preimage of a metrizable space.

A continuous mapping $f:X\rightarrow Y$ is strongly sequentially perfect if
$f$ is closed such that $f^{-1}(F)$ is sequentially compact for each
sequentially compact $F$ in $Y$. A point $x\in X$ is called a strong $q$-point
of a space $X$ if there exists a sequence $\\{U_{n}:n\in\omega\\}$ of open
neighborhoods of $x$ in $X$ such that any sequence $\\{x_{n}:n\in\omega\\}$ of
points in $X$ such that $x_{n}\in U_{n}$ for each $n\in\omega$ has a
convergent subsequence. A space $X$ is said to be a strong $q$-space if every
point of it is a strong $q$-point [10].

Let $A$ be a subset of a space $X$ and $\gamma=\\{U_{n}:n\in\omega\\}$ be a
sequence of open neighborhoods containing $A$. Recall that $\gamma$ is a
countable neighborhood base at $A$ if for each open set $U$ containing $A$
there is a $U_{n}\in\gamma$ such that $U_{n}\subseteq U$. $\gamma$ is called a
strongly countable neighborhood base at $A$, if $\gamma$ is a countable
neighborhood base at $A$ such that each sequence $\\{x_{n}:n\in\omega\\}$ with
$x_{n}\in U_{n}$ for each $U_{n}\in\gamma$ has a convergent subsequence.

######  Lemma 4.8.

Let $f:X\rightarrow Y$ be a continuous and open mapping. Then $f(A)$ has a
strongly countable neighborhood base for each subset $A\subseteq X$ with a
strongly countable neighborhood base.

###### Proof.

Let $\\{W_{n}:n\in\omega\\}$ be a strongly countable neighborhood base of $A$.
Then one can easily show that $\\{f(W_{n}):n\in\omega\\}$ is a strongly
countable open neighborhood base of $f(A)$, because $f$ is a continuous and
open mapping. Take a sequence $\\{y_{n}:n\in\omega\\}$ with $y_{n}\in
f(W_{n})$ for each $n\in\omega$. For each $y_{n}$, take an $x_{n}\in W_{n}$
such that $f(x_{n})=y_{n}$. Then the sequence $\\{x_{n}:n\in\omega\\}$ has a
convergent subsequence $\\{x_{n_{i}}:i\in\omega\\}$, because
$\\{W_{n}:n\in\omega\\}$ is a strongly countable open neighborhood base of
$A$. Then, by the continuity of $f$, one can obtain that
$\\{f(x_{n_{i}}):i\in\omega\\}$ is convergent subsequence of
$\\{y_{n}:n\in\omega\\}$. This finishes the proof. ∎

######  Theorem 4.9.

Suppose that $G$ is a topological group with a strong $q$-point and $H$ is a
closed subgroup of $G$. Then the quotient space $G/H$ is an open and strongly
sequentially perfect preimage of a metrizable space.

###### Proof.

By the homogeneity of $G$, we can assume that the identity $e$ is a strong
$q$-point. Then there is a sequence $\\{U_{n}:n\in\omega\\}$ of open
neighborhoods of $e$ in $G$ satisfies that any sequence
$\\{x_{n}:n\in\omega\\}$ of points such that $x_{n}\in U_{n}$ for each
$n\in\omega$ has a convergent subsequence. We define by induction a sequence
$\\{V_{n}:n\in\omega\\}$ of symmetric open neighborhoods of $e$ in $G$
satisfying the following conditions:

  1. (1)

$V_{0}\subseteq U_{0}$;

  2. (2)

$V_{n+1}^{2}\subseteq V_{n}$ and $V_{n+1}\subseteq U_{n+1}$ for each
$n\in\omega$.

Since $V_{n+1}^{2}\subseteq V_{n}$ and $V_{n}$ is a symmetric open
neighborhood of $e$ for each $n\in\omega$, we have
$\overline{V_{n+1}}\subseteq V_{n}$. One can easily show that the sequence
$\\{V_{n}:n\in\omega\\}$ satisfies the condition (1) in Lemma 4.2. Put
$K=\bigcap_{n\in\omega}V_{n}$. Then $K$ is a closed sequentially compact
subgroup in $G$ such that the sequence $\gamma=\\{V_{n}:n\in\omega\\}$ is a
countable neighborhood base at $K$ in $G$ by Lemma 4.2. Particularly, $\gamma$
is a strongly countable neighborhood base at $K$ in $G$. Thus, by Lemma 4.1,
$K$ is a strongly neutral subgroup in $G$ and the quotient space $K\setminus
G$ is metrizable.

Let $\pi:G\rightarrow G/H$ and $q:G\rightarrow K\backslash G/H$ be natural
quotient mappings. Define a natural mapping $f:G/H\rightarrow K\backslash G/H$
by $f(\pi(x))=q(x)$ for each $x\in G$.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{G/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\backslash
G/H}$

Since $K$ is strongly neutral and $K\setminus G$ is metrizable, $K\backslash
G/H$ is metrizable by Proposition 3.7. By Proposition 3.6, $f$ is a
continuous, open and closed mapping from $G/H$ onto the metrizable space
$K\backslash G/H$.

To show that $f$ is strongly sequentially perfect it is enough to prove that
$f^{-1}(F)$ is sequentially compact for each sequentially compact set
$F\subseteq K\backslash G/H$. Indeed, take arbitrary sequence
$\\{x_{n}:n\in\omega\\}\subseteq f^{-1}(F)$. Then there is a subsequence of
$\\{f(x_{n}):n\in\omega\\}$ converging to some point $z_{0}\in F$, because the
sequence $\\{f(x_{n}):n\in\omega\\}$ is contained in the sequentially compact
set $F$. Without loss of generality, we assume that
$\\{f(x_{n}):n\in\omega\\}$ converges to the point $z_{0}$. Take a $y\in G$
such that $q(y)=z_{0}$. Clearly, $Ky$ has a strongly countable neighborhood
base, because $K$ has a strongly countable open neighborhoods base. Since
$\pi$ is the natural quotient mapping, $\pi$ is continuous and open. By Lemma
4.8, $f^{-1}(z_{0})=\pi(q^{-1}(q(y)))=\pi(KyH)=\pi(Ky)$ as a continuous open
image of the set $Ky$ has a strongly countable neighborhood base
$\\{W_{n}:n\in\omega\\}$. Since $f$ is closed and $f^{-1}(z_{0})\subseteq
W_{n}$ for each $n\in\omega$, we can assume that $f^{-1}(f(W_{n}))=W_{n}$ for
each $n\in\omega$. Then we claim that $W_{i}$ contains infinite elements of
the sequence $\\{x_{n}:n\in\omega\\}$ for each $i\in\omega$. Indeed, if not,
there is an $i_{0}\in\omega$ such that $f(W_{i_{0}})$ contains finite elements
of the sequence $\\{f(x_{n}):n\in\omega\\}$. This contradicts with the
sequence $\\{f(x_{n}):n\in\omega\\}$ converging to the point $z_{0}$, because
$f(W_{i_{0}})$ as an open image of the open set $W_{i_{0}}$ is an open
neighborhood of $z_{0}$. Thus one can take a subsequence
$\\{x_{n_{i}}:i\in\omega\\}$ of the sequence $\\{x_{n}:n\in\omega\\}$ such
that $x_{n_{i}}\in W_{i}$ for each $i\in\omega$. Since
$\\{W_{n}:n\in\omega\\}$ is a strongly countable neighborhood base at
$f^{-1}(z_{0})$, the sequence $\\{x_{n_{i}}:i\in\omega\\}$ has a convergent
subsequence. This implies that the sequence $\\{x_{n}:n\in\omega\\}$ contains
a convergent subsequence, because $\\{x_{n_{i}}:i\in\omega\\}$ is a
subsequence of $\\{x_{n}:n\in\omega\\}$. Let $\\{x_{n_{k}}:k\in\omega\\}$ be a
subsequence of $\\{x_{n}:n\in\omega\\}$ such that it converges to a point
$x_{0}\in G/H$ and $x_{n_{k}}\in W_{k}$ for each $k\in\omega$. Since $H$ is
closed, the quotient space $G/H$ is a Hausdorff space. Thus one can easily
show that $f^{-1}(z_{0})=\bigcap_{n\in\omega}W_{n}$, because the sequence
$\\{W_{n}:n\in\omega\\}$ is a neighborhood base at $f^{-1}(z_{0})$. Thus we
claim that $x_{0}\in f^{-1}(z_{0})\subseteq f^{-1}(F)$. If not, the set
$f^{-1}(z_{0})$ contains finite elements of the sequence
$\\{x_{n_{k}}:k\in\omega\\}$. Without loss generality, we assume that
$\\{x_{n_{k}}:k\in\omega\\}\cap f^{-1}(z_{0})=\emptyset$. Then
$(G/H)\setminus(\\{x_{n_{k}}:k\in\omega\\}\cup\\{x_{0}\\})$ is an open
neighborhood of $f^{-1}(z_{0})$ such that contains no $W_{n}$ for each
$n\in\omega$. This is a contradiction with $\\{W_{n}:n\in\omega\\}$ being a
neighborhood base at $f^{-1}(z_{0})$. We have proved that $f^{-1}(F)$ is
sequentially compact for each sequentially compact set $F\subseteq K\backslash
G/H$. ∎

######  Proposition 4.10.

Let $f:X\rightarrow Y$ be an open and strongly sequentially perfect mapping.
Then $X$ is a strong $q$-space if and only if so is $Y$.

###### Proof.

Assume that $X$ is a strong $q$-space. Take a point $x\in X$ and a strong
$q$-sequence $\\{U_{n}:n\in\omega\\}$ at $x$. Then one can easily show that
$\\{f(U_{n}):n\in\omega\\}$ is a strong $q$-sequence at $f(x)$, because $f$ is
continuous and open. This shows that $Y$ is a strong $q$-space.

Conversely, take a point $x\in X$ and a strong $q$-sequence
$\\{U_{n}:n\in\omega\\}$ at $f(x)$. We claim that the family
$\\{f^{-1}(U_{n}):n\in\omega\\}$ is a strong $q$-sequence at $x$, which
implies that $X$ is a strong $q$-space. Indeed, take $x_{n}\in f^{-1}(U_{n})$
for each $n\in\omega$. Then $f(x_{n})\in U_{n}$ for each $n\in\omega$. Since
$\\{U_{n}:n\in\omega\\}$ is a strong $q$-sequence at $f(x)$,
$\\{f(x_{n})\\}_{n\in\omega}$ has a convergent subsequence
$\\{f(x_{n_{i}})\\}_{i\in\omega}$. Put
$F=\\{f(x_{n_{i}}):i\in\omega\\}\cup\\{y_{0}\\}$, where $y_{0}$ is a
convergent point of the sequence $\\{f(x_{n_{i}})\\}_{i\in\omega}$. Then
$f^{-1}(F)$ is a sequentially compact set in $X$, because $f$ is a strongly
sequentially perfect mapping. Clearly, the subsequence
$\\{x_{n_{i}}\\}_{i\in\omega}$ of the sequence $\\{x_{n}\\}_{n\in\omega}$ is
contained in $f^{-1}(F)$, so $\\{x_{n_{i}}\\}_{i\in\omega}$ has a convergent
subsequence, which is also a convergent subsequence of the sequence
$\\{x_{n}\\}_{n\in\omega}$. This shows that the family
$\\{f^{-1}(U_{n}):n\in\omega\\}$ is a strong $q$-sequence at $x$. ∎

From Theorem 4.9 and Proposition 4.10 it follows that:

######  Corollary 4.11.

[5, Theorem 2.5] A topological group $G$ is a strong $q$-space if and only if
$G$ is an open and strongly sequentially perfect preimage of a metrizable.

######  Proposition 4.12.

Let $(X,d)$ be a pseudometric space. Then the natural mapping $p:X\rightarrow
X/d$ is open and satisfies that $p^{-1}(F)$ is $r$-pseudocompact in $X$ for
each $r$-pseudocompact set $F$ in $X/d$, where the topology on $X$ is induced
by $d$ and $X/d$ is the equivalent classes by $d$, which is endowed with the
quotient topology.

###### Proof.

∎

A point $x\in X$ is called pseudocompacness point of $X$ if there exists a
sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $x$ in $X$
satisfying the condition: (pp) every sequence $\\{V_{n}:n\in\omega\\}$ of non-
empty open sets in $X$ such that $V_{n}\subseteq U_{n}$ for each $n\in\omega$
has a point of accumulation in $X$. A space is said to be pointwise
pseudocompact [1] if each point of $X$ is a pseudocompacness point.

Recall that a subset $A$ of $X$ is said to be a relative pseudocompact, for
brevity, $r$-pseudocompact [4] in $X$ if every infinite family $\gamma$ of
open sets in $X$ such that $U\cap A\neq\emptyset$, for each $U\in\gamma$, has
an accumulation point in $A$.

######  Theorem 4.13.

Suppose that $G$ is a topological group with a pseudocompact point and $H$ is
a closed subgroup of $G$. Then there is a metrizable space $M$ and a
continuous open mapping $f$ from $G/H$ onto $M$ such that $f^{-1}(F)$ is
$r$-pseudocompact in $G/H$ for each $r$-pseudocompact set $F$ in $M$.

###### Proof.

By the homogeneity of $G$, we can assume that the identity $e$ is a
pseudocompact point. Then there is a sequence $\\{U_{n}:n\in\omega\\}$ of open
neighborhoods of $e$ in $G$ satisfying the condition: (pp) every sequence
$\\{W_{n}:n\in\omega\\}$ of non-empty open sets in $G$ such that
$W_{n}\subseteq U_{n}$ for each $n\in\omega$ has a point of accumulation in
$G$. We define by induction a sequence $\\{V_{n}:n\in\omega\\}$ of symmetric
open neighborhoods of $e$ in $G$ satisfying the following conditions:

  1. (1)

$V_{0}\subseteq U_{0}$;

  2. (2)

$V_{n+1}^{2}\subseteq V_{n}$ and $V_{n+1}\subseteq U_{n+1}$ for each
$n\in\omega$.

Put $K=\bigcap_{n\in\omega}V_{n}$. Since $V_{n+1}^{2}\subseteq V_{n}$ and
$V_{n}$ is a symmetric open neighborhood of $e$ for each $n\in\omega$, one can
easily show that $K$ is a closed $r$-pseudocompact subgroup in $G$.

Let $q:G\rightarrow K\backslash G$ and $\pi:G\rightarrow K\backslash G/H$ are
the natural quotient mappings. Then there is a continuous open mapping
$q_{K}:K\backslash G\rightarrow K\backslash G/H$ defined by
$q_{K}(q(x))=\pi(x)$ for each $x\in G$, because $\pi$ and $q$ are continuous
and open by Proposition 3.3.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\pi}$$\textstyle{K\backslash
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{K}}$$\textstyle{K\backslash
G/H}$

In the proof of (1)$\Rightarrow$ (2) in [5, Theorem 2.9], it has proved that
$K\backslash G$ is metrizable. Since $K$ is an $r$-pseudocompact subgroup, $K$
is a bounded subgroup in $G$ by [4, Proposition 2.1]. It follows that $K$ is
precompact from the fact that every bounded subset of a topological group is
precompact [1, Proposition 6.10.2]. By [5, Lemma 2.8] , $K$ is a neutral
subgroup. Thus, from Proposition 3.7 it follows that $K\backslash G/H$ is
pseudometrizable.

Let $\pi_{1}:G\rightarrow G/H$ and $p:G\rightarrow K\backslash G/H$ be natural
quotient mappings. Define a natural mapping $f:G/H\rightarrow K\backslash G/H$
by $f(\pi_{1}(x))=p(x)$ for each $x\in G$.

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{p}$$\textstyle{G/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\backslash
G/H}$

By Proposition 4.12, it is enough to show that $f$ is open and satisfies that
$f^{-1}(F)$ is $r$-pseudocompact in $G/H$ for each $r$-pseudocompact set $F$
in $K\backslash G/H$.

Clearly, $f$ is open. Suppose that the infinite family
$\gamma=\\{O_{n}:n\in\omega\\}$ of open sets in $G/H$ such that $O_{n}\cap
f^{-1}(F)\neq\emptyset$, for each $O_{n}\in\gamma$. Then the infinite family
$\gamma_{1}=\\{f(O_{n}):n\in\omega\\}$ of open sets in $K\backslash G/H$ has
an accumulation point $z_{0}$ in $F$, because $F$ is $r$-pseudocompact in
$K\backslash G/H$. Choose a point $x_{0}$ in $G$ such that $p(x_{0})=z_{0}$.
We claim that $\pi_{1}(KV_{n}x_{0})$ joints with infinite elements in $\gamma$
for each $n\in\omega$. Indeed, if not, there is $\pi_{1}(KV_{n_{0}}x_{0})$
joints with only finite elements in $\gamma$. Then
$f(\pi_{1}(KV_{n_{0}}x_{0}))$ is an open neighborhood of $z_{0}$ such that
$f(\pi_{1}(KV_{n_{0}}x_{0}))$ joints with only finite elements in
$\gamma_{1}$, because

$\displaystyle f^{-1}(f(\pi_{1}(KV_{n}x_{0})))$
$\displaystyle=f^{-1}(p(KV_{n}x_{0}))$
$\displaystyle=\pi_{1}(p^{-1}(p(KV_{n}x_{0})))$
$\displaystyle=\pi_{1}(KV_{n}x_{0}H)$ $\displaystyle=\pi_{1}(KV_{n}x_{0})$

holds for each $n\in\omega$. This is a contradiction with $z_{0}$ being an
accumulation point of $\gamma_{1}$. Without loss of generality, we assume that
$\pi_{1}(KV_{n}x_{0})\cap O_{n}\neq\emptyset$ holds for each $n\in\omega$. One
can easily show that the family $\\{KV_{n}:n\in\omega\\}$ satisfies the
condition (pp), so does the family $\\{KV_{n}x_{0}:n\in\omega\\}$. Thus the
family $\\{KV_{n}x_{0}\cap\pi_{1}^{-1}(O_{n}):n\in\omega\\}$ has an
accumulation point $y_{0}$ in $G$, because
$KV_{n}x_{0}\cap\pi_{1}^{-1}(O_{n})$ is open and contained in $KV_{n}x_{0}$
for each $n\in\omega$. We claim that $y_{0}\in Kx_{0}$. Indeed, since

$\overline{KV_{n+2}x_{0}}\subseteq\overline{V_{n+2}^{2}x_{0}}\subseteq\overline{V_{n+1}x_{0}}\subseteq
V_{n+1}V_{n+1}x_{0}\subseteq V_{n}x_{0}$

holds for each $n\in\omega$, we have that

$y_{0}\in\bigcap_{n\in\omega}\overline{KV_{n}x_{0}}\subseteq\bigcap_{n\in\omega}V_{n}x_{0}=(\bigcap_{n\in\omega}V_{n})x_{0}=Kx_{0}.$

Thus, $\pi_{1}(y_{0})$ is an accumulation point of the family
$\\{\pi_{1}(KV_{n}x_{0})\cap O_{n}:n\in\omega\\}$ contained in
$\pi_{1}(Kx_{0})$. Since, by $z_{0}\in F$,

$f^{-1}(z_{0})=\pi_{1}(p^{-1}(z_{0}))=\pi_{1}(Kx_{0}H)=\pi_{1}(Kx_{0})\subseteq
f^{-1}(F),$

we have that $\pi_{1}(y_{0})\in f^{-1}(F)$. Thus we have proved that the
family $\\{O_{n}:n\in\omega\\}$ has an accumulation point $\pi_{1}(y_{0})$
contained in $f^{-1}(F)$. ∎

######  Proposition 4.14.

Let $f$ be a continuous open mapping from $X$ onto a first-countable space $Y$
such that $f^{-1}(F)$ is $r$-pseudocompact in $X$ for each $r$-pseudocompact
set $F$ in $Y$. Then $X$ is a pointwise pseudocompact-space.

###### Proof.

Take a point $x\in X$ and assume that a decreasing family
$\\{U_{n}:n\in\omega\\}$ is a base at $f(x)$. We claim that the family
$\\{f^{-1}(U_{n}):n\in\omega\\}$ satisfies the condition (pp): every sequence
$\\{V_{n}:n\in\omega\\}$ of non-empty open sets in $X$ such that
$V_{n}\subseteq f^{-1}(U_{n})$ for each $n\in\omega$ has a point of
accumulation in $X$. Clearly, $f(x)$ is an accumulation of the family
$\\{f(V_{n}):n\in\omega\\}$. Then one can find a sequence
$\\{y_{n}\\}_{n\in\omega}$ in $Y$ such that the sequence
$\\{y_{n}\\}_{n\in\omega}$ converges to the point $f(x)$ and joints with
infinitely elements of the family $\\{f(V_{n}):n\in\omega\\}$. Put
$F=\\{y_{n}:n\in\omega\\}\cup\\{f(x)\\}$. Clearly, the set $F$ is
$r$-pseudocompact in $Y$, so $f^{-1}(F)$ is an $r$-pseudocompact set in $X$
such that $f^{-1}(F)$ joints with infinitely elements of the family
$\\{V_{n}:n\in\omega\\}$. Thus, there is an accumulation of the family
$\\{V_{n}:n\in\omega\\}$ in $f^{-1}(F)$. This completes the proof. ∎

From Theorem 4.13 and Proposition 4.14 it follows the following result:

######  Corollary 4.15.

[5, Theorem 2.9] A topological group $G$ is a pointwise pseudocompact-space if
and only if there is a continuous open mapping $f$ from $G$ onto a metrizable
space $M$ such that $f^{-1}(F)$ is a $r$-pseudocompact set in $G$ for each
$r$-pseudocompact set $F$ in $M$.

##  5 Other related results

######  Theorem 5.1.

Let $H$ be a closed neutral subgroup of a topological group $G$ such that $H$
contains a subgroup $K$ such that the quotient space $G/K$ is first-countable.
Then the quotient space $G/H$ is metrizable.

###### Proof.

Since $H$ is a closed neutral subgroup, by (2) in Remark 2 the topology on the
quotient space $G/H$ is induced by the separated uniformity
$\mathcal{U}_{G/H}^{l}$ and $\mathscr{B}_{G/H}^{l}$ is a base for
$\mathcal{U}_{G/H}^{l}$. Thus, to show that the quotient space $G/H$ is
metrizable, it is only to show that $\mathcal{U}_{G/H}^{l}$ has a countable
base by Theorem 2.3. Let $q:G\rightarrow G/K$ and $\pi:G\rightarrow G/H$ are
the natural quotient mapping. Since the quotient space $G/K$ is first-
countable, there is a countable family $\gamma=\\{O_{n}:n\in\omega\\}$ of
symmetric open neighborhoods of the identity $e$ in $G$ such that
$\\{q(O_{n}):n\in\omega\\}$ is a base at $q(e)$. We claim that every element
of $\mathscr{B}_{G/H}^{l}$ contains an element of
$\\{E_{O_{n}}^{l}:O_{n}\in\gamma\\}$, where
$E_{O_{n}}^{l}=\\{(\pi(x),\pi(y):)\pi(y)\in\pi(xO_{n})\\}$ for all $n$. This
implies that $\mathcal{U}_{G/H}^{l}$ has a countable base, because
$\mathscr{B}_{G/H}^{l}$ is a base for $\mathcal{U}_{G/H}^{l}$.

Indeed, take arbitrary $E_{V}^{l}\in\mathscr{B}_{G/H}^{l}$, where
$E_{V}^{l}=\\{(\pi(x),\pi(y):)\pi(y)\in\pi(xV)\\}$ and
$V\in\mathscr{N}_{s}(e)$. Then there is $O_{n}\in\gamma$ such that
$q(O_{n})\subseteq q(V)$, which implies that $O_{n}K\subseteq VK$. We claim
that $E_{O_{n}}^{L}\subseteq E_{V}^{l}$. Take any $(\pi(x),\pi(y))\in
E_{O_{n}}^{L}$, which implies that $\pi(y)\in\pi(xO_{n})$. Thus, we have that
$y\in yH\subseteq xO_{n}H$. Since we have proved that $O_{n}K\subseteq VK$, we
have that $xO_{n}K\subseteq xVK$, which implies that $xO_{n}KH\subseteq xVKH$.
This implies that $xO_{n}H\subseteq xVH$ by $K\subseteq H$. Therefore, we have
that $y\in xO_{n}H\subseteq xVH$, which implies that
$\pi(y)\in\pi(xVH)=\pi(xV)$. Thus $(\pi(x),\pi(y))\in E_{V}^{l}$. ∎

Since the quotient space $H\backslash G$ and $G/H$ are homeomorphic for every
subgroup $H$ in a topological group $G$, from Theorem 5.1 it follows:

######  Corollary 5.2.

[3, Theorem 3.1] Let $H$ be a closed neutral subgroup of a topological group
$G$. If the space $H\backslash G$ is first-countable, then it is metrizable.

######  Corollary 5.3.

Let $H$ be a closed neutral subgroup of a pointwise pseudocompact topological
group $G$. Then the quotient space $G/H$ is metrizable if and only if $G/H$
has a $G_{\delta}$-point.

######  Corollary 5.4.

Let $H$ be a closed neutral subgroup of a topological group $G$ with a
q-point. Then the quotient space $G/H$ is metrizable if and only if $G/H$ has
a $G_{\delta}$-point.

In 2021, X. Ling et al. proved that: Let $H$ be a closed subgroup of a
feathered group $G$; then the quotient space $G/H$ is metrizable if and only
if $G/H$ is first-countable [7, Theorem 4.4]. However, the proof seems to have
a gap, because there is a separated uniformity with no countable base but it
induces a metrizable topology (see [2, Example 8.1.7]). From Remarks 2 and 1
it follows that $\mathcal{U}_{G/K}^{r}$ and $\mathcal{U}_{G/K}^{l}$ are
separated uniformities on the quotient space $G/K$ which induce the topology
of $G/K$ in [7, Theorem 4.4], but $\mathcal{U}_{G/K}^{r}$ may be have no
countable base. From Corollary 5.4 we have the following result:

######  Corollary 5.5.

Let $H$ be a closed neutral subgroup of a feathered group $G$. Then the
quotient space $G/H$ is metrizable if and only if $G/H$ has a
$G_{\delta}$-point..

## References

## References

  * [1] A.V. Arhangel’skii, M.G. Tkachenko,  _Topological Groups and Related Structures_ ,  Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
  * [2] R. Engelking,  _General Topology,_ Sigma Series in Pure Mathematics, Heldermann, Berlin, 1989.
  * [3] M. Fernández, I. Sánchez, M. Tkachenko,  _Coset spaces and cardinal invariants_ ,  Acta Math. Hung. 2 (159) (2019), 486-502.
  * [4] S. Hernández, M. Sanchis, M. G. Tkachenko,  _Bounded sets in spaces and topological groups_ ,  Topol. Appl. 101 (2000), 21-43.
  * [5] S. Lin, L.-H. Xie, D. Chen,  _Some generalized countably compact properties in topological groups_ ,  Topol. Appl. 339 (2023) 108705.
  * [6] S. Lin, Z. Yun,  _Generalized Metric Spaces and Mappings_ ,  Atlantis Studies in Mathematics, vol. 6, Atlantis Press, Paris, 2016.
  * [7] X. Ling, W. He, S. Lin,  _Cardinal invariants of coset spaces,_ Topol. Appl. 299 (2021), 107735.
  * [8] E. Michael,  _A quintuple quotient quest_ ,  Gen. Topol. and Appl. 2:1 (1972), 91-138.
  * [9] K. Morita,  _Product of normal space with metric spaces,_ Math. Ann. 154 (1966), 365-382.
  * [10] L.X. Peng, Y. Liu,  _Topological groups with a (strongly) $q$-point_,  Houston J. Math. 2 (47) (2021), 499-516.
  * [11] J. Poncet,  _Une classe d’espaces homog‘enes poss edant une mesure invariante_ ,  C.R.Acad. Sci. Paris, 238(1954), 553-554.


# Atrial Fibrillation Detection Using RR-Intervals for Application in
Photoplethysmographs

Georgia Smith  The University of Virginia’s College at Wise  Dr. Yishi Wang
The University of North Carolina at Wilmington

###### Abstract

Atrial Fibrillation is a common form of irregular heart rhythm that can be
very dangerous. Our primary goal is to analyze Atrial Fibrillation data within
ECGs to develop a model based only on RR-Intervals, or the length between
heart-beats, to create a real time classification model for Atrial
Fibrillation to be implemented in common heart-rate monitors on the market
today. Physionet’s MIT-BIH Atrial Fibrillation Database [5] and 2017 Challenge
Database [1] were used to identify patterns of Atrial Fibrillation and test
classification models on. These two datasets are very different. The MIT-BIH
database contains long samples taken with a medical grade device, which is not
useful for simulating a consumer device, but is useful for Atrial Fibrillation
pattern detection. The 2017 Challenge database includes short ($<60sec$)
samples taken with a portable device and reveals many of the challenges of
Atrial Fibrillation classification in a real-time device. We developed
multiple SVM models with three sets of extracted features as predictor
variables which gave us moderately high accuracies with low computational
intensity. With robust filtering techniques already applied in many
Photoplethysmograph-based consumer heart-rate monitors, this method can be
used to develop a reliable real time model for Atrial Fibrillation detection
in consumer-grade heart-rate monitors.
_Keywords:_ Atrial Fibrillation Classification, Support Vector Machine,
Feature Extraction, Real Time Model, Physionet, Electrocardiogram,
Photoplethysmogram, MIT-BIH, Physionet, Consumer Implementation, RR-Intervals,
Transition Matrix

##  1 Introduction

Atrial Fibrillation (AFib) is the most common form of Arrhythmia, or irregular
heart rhythm. It can be very dangerous, causing an increased risk of stroke
and heart attack/failure [8]. Symptoms of Atrial Fibrillation include
increased heart rate, heart palpitations, and low blood pressure. Atrial
Fibrillation is characterized by two main features: an absence of P-waves in
an Electrocardiogram (ECG) Recording, and “Irregular Irregularity,” meaning
that there is no pattern to when AFib rhythms start or stop and no specific
pattern to the heart rhythm. The irregularity between heartbeat in Atrial
Fibrillation is caused by abnormal electrical activity in the atrial, or top,
chambers of the heart.

The P-Wave, which characterizes Atrial Fibrillation, has a low amplitude in
comparison to the R-Wave which is the peak of ECG recordings. We chose to use
the interval between adjacent R-wave peaks (RR-Interval) to detect Atrial
Fibrillation because P-Waves are often misidentified due to noise within
Electrocardiogram recordings and because in Photoplethysmographs (PPGs), which
are used in consumer heart-rate monitors, just the heart beat is identified.
To analyze and identify RR-Interval patterns of ECG recordings of individuals
experiencing AFib versus Normal heart-beat rhythms we used the MIT-BIH Atrial
Fibrillation Database [5] and the Physiobank 2017 Challenge Dataset [1]. The
MIT-BIH Database includes 230 hours of Electrocardiogram (ECG) recordings with
Atrial Fibrillation labelled, which was used to learn about AFib patterns and
test potential models on. The MIT-BIH Database has been used by many authors
to detect Atrial Fibrillation due to its robust and longitudinal nature [9, 3,
6, 7, 4]. The MIT-BIH Database has also been thoroughly cleaned for accurate
model building. The 2017 Challenge Dataset is less thoroughly cleaned set of
recordings which allowed us to test the robustness of models.

We first reviewed and considered other research on Atrial Fibrillation
classification from different academics. We then focused on initial analysis
of the data where we extracted and cleaned the data from the MIT-BIH Atrial
Fibrillation Database and the 2017 Challenge Dataset, identifying patterns of
AFib in the ECG recordings, and developing potential modelling techniques to
detect dangerous levels of Atrial Fibrillation. Once the data was extracted we
developed multiple features and then subset the data into 45-second intervals,
including the extracted features. We then used these samples to test
classification models.

###  1.1 Goal

Atrial Fibrillation is difficult to diagnose because of its irregular nature.
Often diagnosis of Atrial Fibrillation is incidental, which is not ideal given
AFib can increase the risk of stroke in adults older than 45 by 20% [7]. Our
goal is to address this by developing a simple, accurate model for real time
Atrial Fibrillation detection using common heart-rate monitors on the market
today (Apple Watch, FitBit, etc.). These aforementioned heart-rate monitors
often do not use Electrocardiograms, but instead use Photoplethysmograms
(PPGs), which measure the expansion and contraction of blood vessels as the
heart pumps blood throughout the body. This technology is cheaper and easier
for day to day use because it can be placed anywhere that there is a blood
vessel. While application in PPGs is the goal, we are using ECG data to
identify patterns of AFib through RR-intervals to simulate PPG signals because
PPG data from Atrial Fibrillation patients is not publicly accessible.

###  1.2 Contribution

  * •

Cross-database Transfer Learning

  * •

Transition Matrix Predictor Variables

  * •

$RR_{100}$ modification to $RR_{200}$

  * •

Outlier Removal

  * •

$RR_{var}$ modifications

  * •

Implementation for Electrocardiograms and Photoplethysmographs

  * •

RR-Interval-based Predictor Variables

##  2 Literature Review

Electrocardiograms provide immense amount of data to work with in identifying
Atrial Fibrillation and, with Physiobank’s open source Electrocardiogram
databases, research into automated classification of Atrial Fibrillation and
other heart conditions is very thorough. While there is much literature out
there for Atrial Fibrillation, our main focus is on extraction of features
based on RR-intervals with low computational intensity for implementation in a
real time PPG/ECG device.

###  2.1 Moody and Mark

Moody and Mark, in their 1983 paper, attempted to predict future heart-beat
rhythms given previous heart-beats in order to identify patterns of Atrial
Fibrillation. They did this by creating a running mean and classifying the
individual RR-Intervals as long, normal, or short by comparison to this
running mean [3].

$rmean_{i}=0.75*rmean_{i-1}+0.25*RR_{i},$ (1)
$C_{i}=\begin{cases}short&\text{if }RR_{i}<0.85\times rrmean_{i}\\\
long&\text{if }RR_{i}>1.15\times rrmean_{i}\\\
normal&\text{otherwise}\end{cases}.$ (2)

They then used the classifications to develop transition matrices for a Markov
model to predict the next heart beat rhythm. While the prediction of the next
rhythm is not our concern, the transition matrices Moody and Mark utilized
allow a quantitative measurement for how much the heart-beat varies from a
“normal” heart rhythm.

###  2.2 Rubio and Beregaña

In their 2011 paper, Rubio and Beregaña [7] compared 9 different extraction
and classification techniques used for Atrial Fibrillation classification.
First they filtered their ECG data using a high pass filter to remove Baseline
Wander, a common noise in ECG data due to patient movement and incorrect ECG
node placement. They then set a maximum and minimum threshold for reading
R-Peaks to reduce incorrect R-Peak detection due to noise. The cleaning
methods were very simple, yet allowed for more accurate readings even in the
presence of noise. The simplicity of these cleaning methods could be very
useful in implementation of a real-time AFib-Detection device. Once Rubio and
Beregaña cleaned the ECG data they tested 9 extraction techniques. Out of the
9 extracted measurements tested we identified two that may be useful: Ghodrati
et al [4] and Young et al [9].

Ghodrati et al used a similar algorithm to Moody and Mark, except instead of
classifying the RR-Intervals into transitions matrices they developed an RR-
variance algorithm defined by

$RRvar_{i}=\frac{|RR_{i}-RR_{i-1}|}{rmean_{i}},$ (3)

and $\overline{RR_{l}}$ is defined recursively by

$rmean_{i}=0.9*\overline{RR_{i-1}}+0.1*RR_{i}.$ (4)

Instead of quantifying how much a heart-beat varies from “normal” with
transitions, Ghodrati et al’s $RRvar$ gives a longitudinal measurement similar
to a running mean which quantifies how much RR-Intervals are changing.

Young et al [9] used a much simpler classification technique using no
statistical models, but instead a simple classification algorithm using
counting defined as the $RR_{100}$ algorithm.

$\displaystyle\begin{split}RR_{i}-RR_{i-1}>100ms\rightarrow count=count+1,\\\
RR_{i}-RR_{i-1}<100ms\rightarrow count=count-1,\end{split}$ (5)
$count>6\rightarrow AFib.$ (6)

We identified the $count$ variable, in Equations 5 and 6, as a potential
covariate for our model.

##  3 Data

###  3.1 The MIT-BIH Database

The MIT-BIH Atrial Fibrillation Database includes 23, complete, 10-Hour ECG
Recordings from individuals diagnosed with Atrial Fibrillation. The dataset is
oversampled with AFib data, which is useful in building a model, but not for
an accurate testing set. The dataset includes four rhythm types, Atrial
Fibrillation (AFib), Normal (N), Atrial Flutter (AFL), and AV Junctional
Rhythm (J).

|  AFib | AFL | J | N
---|---|---|---|---
Proportion of Data | 0.398 | 0.006 | 0.004 | 0.592
Table 1: Proportion of Different Heart-Rhythms in MIT-BIH Data

The recordings include RR-Intervals, annotations sectioning heart-beats into
categories, and digitized signals for each subject. The readings were taken at
250 samples per second to increase their reliability. On average, the
recordings include 48,000 heart-beats. The data in the MIT-BIH Atrial
Fibrillation Database is subset into three files: reference annotation files,
beat annotation files, and digitized signal files. For this analysis we
focused on the reference annotation files (.atr files) and the beat annotation
files (.qrs files) to extract RR-Intervals and annotations.

The MIT-BIH Database is a very reliable Atrial Fibrillation Database that is
often used in the field of Atrial Fibrillation and other heart-rhythm based
research [7, 6, 3]. This dataset is so reliable because it have been cleaned
and tested in multiple research studies. Physionet also used two or more
medical professionals to label the rhythms for every ECG recording.

###  3.2 Physionet 2017 Challenge Data

In 2017 Physionet, the same organization which hosts the MIT-BIH Database,
held a competition for Atrial Fibrillation Classification. A main focus of
this competition was to find methods of Atrial Fibrillation Detection without
over-sampled, clean data and develop methods to identify more than two
classes. Therefore, the data in the Physiobant 2017 Challenge Dataset includes
four classifications: Noise, Atrial Fibrillation, Normal, and Other.

|  Normal | AF |  Other |  Noisy |  Total
---|---|---|---|---|---
Amount | 5,154 | 771 | 2,557 | 46 | 8,528
Table 2: Types of Rhythm within 2017 Competition Data

The Noisy category includes all ECG recordings which are unreadable due to an
excess of noise and the Other category includes all identifiable heart-rhythms
that are not AFib or Normal.The 2017 Challenge dataset also has a range of
times from 9-60 seconds, although most recordings are 30 seconds long. The
varying times cause our samples to be varying in length causing an extra
variable that we most likely would not have to account for in the continuous
model/device which we hope to develop.

Figure 1: Distribution of Sample-Lengths in 2017 Challenge Dataset

In the 2017 Challenge dataset the recordings were taken with a single lead,
portable Electrocardiogram device developed by AliveCor, causing further
problems of R-Peak detection due to noise. Because this data was taken with a
portable device, it is a more accurate test set for future implementation of
AFib classification models.

##  4 Data Preprocessing and Feature Extraction

###  4.1 Extraction of RR-Intervals

Both the MIT-BIH Database and the 2017 Challenge Dataset came in digitized
formats for use with Moody’s Wave-Form-Database (WFDB) [2]. To extract the RR-
Intervals from these files we also used the WFDB of Functions and extraction
methods. On the 2017 Challenge Dataset we first had to extract .dat files from
the digitized matlab files provided using “mat2wfdb.” We then had to extract
.qrs files from the .dat files for the 2017 Challenge Dataset using WFDB’s
“wqrs” function. We then had both the MIT-BIH dataset and the 2017 Challenge
dataset in .qrs form where RR-Intervals could be detected. We first used the
“ann2rr,” or annotations to RR-Intervals, function to extract RR-Intervals
from the .qrs files. The data extracted from this function included the sample
where each heart-beat started and ended, as well as the RR-Intervals in
seconds. Then, to extract the labelled heart-beat rhythms from the MIT-BIH
dataset we used the “rdann,” or read annotations, function to extract heart-
beat annotations for the .atr files. The rhythms for the 2017 Challenge
dataset were labelled in a separate .csv provided by Physionet. The 2017
Challenge data heart-beat rhythms were classified individually by classifying
the entire sample as a single rhythm due to the short recording times.

The data extracted from “rdann” included elapsed time at start, sample number
at start, and annotation. The annotations classified the MIT-BIH heart-beats
into four rhythm categories: Atrial Fibrillation (AFib), Normal (N), Atrial
Flutter (AFL), and AV Junctional Rhythm (J). Because the annotations were
formatted by range they had to be merged the RR-Intervals from the 23 MIT-BIH
datasets. To merge the files we used the sample numbers in the extracted files
to set each group of RR-Intervals as their corresponding annotation.

###  4.2 Signal Processing

When looking at the 2017 Challenge dataset an abnormal amount of noise was
present in comparison to the MIT-BIH database. When extracted the same way as
the MIT-BIH dataset the WFDB [2] extracted less than 8 RR-Intervals for
samples with 30 or more seconds of data due to noise. The noise present is
accounted for because the 2017 Challenge dataset was collected using a
portable, single-lead ECG. The MIT-BIH database was collected using a medical-
grade, dual-lead ECG in a constant setting for accurate recordings. Taking
this into consideration we looked into signal processing methods for the 2017
Challenge dataset.

Figure 2: Subject A01723 Raw Data, with $<8$ RR-Intervals observed by WFDB,
Heart Rhythm Classified as Other

In 2011, Rubio and Beregaña used high pass filters on the MIT-BIH database for
their testing of Atrial Fibrillation Classification Models to reduce noise and
baseline wander. Taking this into consideration we applied low pass and high
pass filtering techniques used in unison as a bandpass filter on the 2017
Challenge Dataset.

Figure 3: Subject A01723 Raw Data Overlayed by Filtered Data

This resulted the noise to be decreased with only R-Peaks and other extreme
noise readings to be peaks on the ECG.

Figure 4: Subject A01723 Filtered Data with R-Peaks Detected

This did not remove all problems though, ECGs with noise at an extreme
amplitude still resulted in incorrect R-Interval readings. To address this we
developed outlier thresholds at $\text{heart-rate}>200bpm$, $\text{heart-
rate}<20bpm$, and $\overline{drr}>1sec$, where $\overline{drr}$ is the
arithmetic mean of the difference in RR-Interval length defined as

$\overline{drr}=\frac{|RR_{i}-RR_{i-1}|}{\Sigma RR}.$ (7)

Any sample from the 2017 Challenge Dataset which was classified by one or more
of the thresholds was deemed an outlier.

Figure 5: Outlier Example, Subject A06006, Heart-Rhythm Classified as Normal

With these outliers specified we were able to further filter just those
recordings that had noise distorting R-Peak identifications by removing
sections with abnormal noise, which were usually isolated at the beginning and
end of every recording. Once we removed those sections we processed the
remaining data with a bandpass filter and identified R-Peaks. We then checked
for outliers once more and found the data to be more reliable in detecting
R-Peaks.

Figure 6: Subject A06006 R-Peak Detection with Noise Removed

###  4.3 Subsetting

Once the RR-Intervals were extracted we removed any RR-Intervals which were
more than four standard deviations away from the mean. This was done because
there are recordings within the MIT-BIH dataset which have bad readings marked
as no heart-beat, resulting in some RR-Intervals being 5-10 seconds. With
outliers removed from both datasets, we applied Moody and Mark’s running mean
for each RR-Interval defined recursively by Equation 1. A running mean was
used because of the longitudinal nature of heart-beat data and the objective
of this analysis to develop a method for real-time AFib detection. This
running mean was also determined to increase accuracy by Moody and Mark. We
then classified each individual RR-Interval as short, normal, or long based
defined by Equation 2. This process done on the MIT-BIH dataset and the 2017
Challenge dataset.

With the RR-Interval Classifications we then developed transition matrices for
each subject. The transition matrices tallied each transition (Short-to-Short,
Short-to-Normal, Short-to-Long, etc.) for future reference.

The complete, merged, data for each subject now included start and stop
samples, RR-Interval in seconds, Annotations (within the MIT-BIH Dataset),
running mean, and classification as short, normal, or long. For the MIT-BIH
Database we subset each ECG recording in 45-second samples for better
simulation of real-time classification. There was no need to subset the 2017
Challenge dataset due to the already short sample lengths. Within each subset
of the MIT-BIH data and recording from the 2017 Challenge data we developed a
transition matrix based on the RR-Interval classification.

|  |  | From |
---|---|---|---|---
|  |  Short |  Normal |  Long
| Short | 8 | 6 | 7
To | Normal | 9 | 14 | 6
| Long | 4 | 10 | 4
Table 3: Example of Transition Matrix of RR-Interval Classification for a
Random 45-sec/68-beat Sample from the MIT-BIH Dataset

With the transition matrices in Table 3, we found the proportions of each
transition.

|  |  | From |
---|---|---|---|---
|  |  Short |  Normal |  Long
| Short | 0.118 | 0.088 | 0.103
To | Normal | 0.132 | 0.206 | 0.088
| Long | 0.059 | 0.147 | 0.059
Table 4: Example of Transition Matrix of RR-Interval Classification
Proportions for 45-sec/68-beat Sample in Table 3

We then unlisted the transmission matrices exemplified in Figure 4 into the
form: [Short-to-Short, Short-to-Normal, Short-to-Long, Normal-to-Short,
Normal-to-Normal, Normal-to-Long, Long-to-Short, Long-to-Normal, Long-to-
Long]. We also utilized Young et al’s $RR_{100}$ algorithm shown in Equation
5, and 6 and Ghodrati et al’s $RR_{var}$ from Equation 3. We modified both
Ghodrati and Young’s measurements to better reflect the data.

Ghodrati et al’s $RR_{var}$ had a running mean defined by Equation 4, which we
changed to reflect Equation 1 for consistency. Young et al’s $RR_{100}$
algorithm had to be altered to reflect the sample. Because we used the
algorithm as a measurement instead of a classification technique we altered
the count to become a proportion for different sample sizes. In a preliminary
study we tested model performances with Young’s 100ms threshold incremented by
10ms. Doing this we found that 200ms was a better threshold for the MIT-BIH
and 2017 Challenge datasets. This new measurement was defined as $RR_{200}$:

$\displaystyle\begin{split}RR_{i}-RR_{i-1}>200ms\rightarrow count=count+1,\\\
RR_{i}-RR_{i-1}<200ms\rightarrow count=count-1,\end{split}$ (8)
$RR_{200}=\frac{count}{\Sigma RR}.$ (9)

We then extracted the length of each subset in beats and seconds, the start
and end sample for each subset, and the majority annotation. We did not
consider all four rhythm categories within the MIT-BIH dataset for this
analysis, instead we dichotomized each 45-second sample as either AFib or not-
AFib. This was done to better identify patterns of just Atrial Fibrillation
heart-beats and was also deemed by Moody and Mark in 1983 not to significantly
affect results. We also did not consider segments that include both AFib and
not-AFib for this study because for this analysis we focus on classifying
entire segments as AFib or not-AFib. We removed the segments with multiple
rhythms for future analysis. With the extracted features we developed a new
dataset of just the extracted sample features for each subject. For the 2017
Challenge data we considered both binary and multi-class classification.

##  5 Analysis

###  5.1 RR-Intervals and Running Mean

We first looked at the distribution of the RR-Intervals and running means for
each subject using the data from the completed merged datasets. Using plots of
the RR-Intervals over the recording period and running means over the
recording period we then highlighted all annotations of identified Atrial
Fibrillation.

Figure 7: Plot of Running Mean (top) and RR-Intervals (bottom) for Subject 04048 from MIT-BIH Dataset, Highlighted in Green when Subject is in AFib Figure 8: Plot of Running Mean (top) and RR-Intervals (bottom) for Subject 07879 from MIT-BIH Dataset, Highlighted in Green when Subject is in AFib |  Subject |
---|---|---
| 04048 | 07879
RR-Int | 0.009 | 0.032
Running Mean | 0.007 | 0.022
Table 5: Variance of RR-Intervals and Running Mean in Subjects 04048 and 07879

Table 5 shows the variances of both the RR-Intervals and the Running mean over
the entire ECG recording for each subject. Even including the sections without
AFib, Subject 07879 shows a much higher variance than Subject 04048. This
signified to us that looking deeper into the transition proportions we subset
previously might also show differences in AFib and not-AFib sections.

###  5.2 Transition Matrices Proportions

Figure 9: Transition Proportions of each 45-second Subset of Subject 04048
From MIT-BIH Data over Entire Recording Period Figure 10: Transition
Proportions of each 45-second Subset of Subject 07879 From MIT-BIH Data over
Entire Recording Period

Once we identified some patterns of subjects in AFib also having high variance
among RR-Intervals and Running Mean, we looked closer at the transition
proportions created. We plotted the density of each transition as well as the
proportions over the whole recording period.

Above, Figures 9 and 10 give the transition proportions extracted over the
entire recording period. Where Atrial Fibrillation occurs there is a rise in
all lines in ‘From Short’ and ‘From Long’ in Figure 10. In ‘From Normal,’ in
Figure 10, the Normal-to-Normal (N-N) proportions begin to converge with N-S
and N-L proportions, while in Figure 9 and all not-AFib portions of 10 N-N is
concentrated near 1 and all other transition proportions are very low. This
distinct pattern was seen in preliminary studies of all 23 Subjects, details
can be found at our website: https://github.com/025georgialynny/afib.

Identifying this pattern in a model does cause some issues because to
continuously read in and store transition proportions to develop plots similar
to those above it would require a significant amount of memory and time to
identify AFib. Because of this problem, we took a deeper investigation into
the transition proportions to develop a better understanding of the
differences between individuals with low AFib (Subject 04048) and those with a
higher amount of AFib (Subject 07879).

Figure 11: Density of Transition Proportions of the 45-second Subsets of
Subject 04048 From the MIT-BIH Dataset over Entire Recording Period Figure 12:
Density of Transition Proportions of the 45-second Subsets of Subject 07879
From the MIT-BIH Dataset over Entire Recording Period

Just looking at the ‘From Normal’ plots in Figures 11 and 12 there is a clear
difference in the spread of the proportions’ density. In Figure 11, the data
is Bernoulli distributed with binary peaks while Figure 12 shows a much more
spread distribution of density of proportions.

##  6 Classification Models

###  6.1 Model and Feature Selection

Once the data was subset and features had been extracted we used the MIT-BIH
dataset to find the best model to use. This was done because the MIT-BIH
dataset is much cleaner giving us a better idea of model performance for AFib-
related ECG data. We built our models with MIT-BIH data and then tested with a
sample of 300 subjects from the 2017 Challenge data to test the robustness of
each model. To do this we dichotomized each sample in the 2017 data as AFib or
not AFib to account for the difference in rhythm categories used by the
datasets. We compared Logistic Regression (LogReg), Linear Discriminant
Analysis (LDA), Quadratic Discriminant Analysis (QDA), Boosting, Support
Vector Machines (SVM) with a Radial Kernel (after preliminary analysis of
multiple kernels), and Random Forest (RF) as models to classify AFib.

|  Accuracy |  Specificity |  Sensitivity
---|---|---|---
LogReg | 85.619% | 90.873% | 57.447%
LDA | 81.605% | 93.304% | 46.667%
QDA | 76.254% | 82.014% | 0.000%
Boosting | 82.274% | 88.583% | 53.333%
SVM | 87.291% | 88.364% | 75.000%
RF | 78.930% | 95.146% | 56.989%
Table 6: Comparison of Models using MIT-BIH Data to Train and 2017 Challenge
to Test

Support Vector Machines, with a radial kernel, outperformed the other models,
especially just looking at the Sensitivity of the models. Because Atrial
Fibrillation is a serious health disorder we are willing to sacrifice
sensitivity for specificity to increase predictions of AFib at a reasonable
rate. With this in mind we continued with SVM with a radial kernel as our
preferred model.

Label |  Key
---|---
Model 1 | 9 Transition Proportions, $RR_{var}$, and $RR_{200}$
Model 2 | Normal-Normal, $RR_{var}$, and $RR_{200}$
Model 3 | Normal-Long, $RR_{var}$, and $RR_{200}$
Confusion Matrix Columns | Reference Classifications
Confusion Matrix Rows | Predicted Classifications
Table 7: Key for Table 8 |  Confusion Matrix |  Accuracy |  Specificity |  Sensitivity
---|---|---|---|---
Model 1 |  |  | N | AF
---|---|---
N | 10,386 | 144
AF | 217 | 6,861
97.948% | 97.965% | 97.944%
Model 2 |  |  | N | AF
---|---|---
N | 9,652 | 300
AF | 951 | 6,705
92.895% | 91.031% | 95.717%
Model 3 |  |  | N | AF
---|---|---
N | 10,285 | 175
AF | 318 | 6,830
97.200% | 97.001% | 97.502%
Table 8: Performance of Models with Different Prediction Variables Confusion Matrix |  Accuracy |  Specificity |  Sensitivity
---|---|---|---
|  | N | AF
---|---|---
N | 10,340 | 263
AF | 195 | 6,810
97.399% | 98.149% | 96.282%
Table 9: LOO-CV on MIT-BIH Data with Normal-to-Long Transition Proportion,
$RR_{200}$, and $RR_{var}$ Performance

A problem with all models is the collinearity of our features, since all of
the transition proportions summed to one. For further model testing in the
future the collinearity will need to be reduced for model validation. To
reduce the collinearity without the computational intensity of dimension
reduction we compared the results using all transition proportions with
$RR_{200}$ and $RR_{var}$ with just Normal-to-Long or Normal-to-Normal with
$RR_{200}$ and $RR_{var}$.

With just Normal-to-Long paired with $RR_{200}$ and $RR_{var}$ as predictor
variables the SVM model performs very close results to all Model 1 This is
very preferable because it prevents collinearity for comparison of SVM with
other models and decreases the computational intensity of the model for future
implementation.

Leave-One-Out Cross-Validation (LOO-CV) performed very similarly with Normal-
to-Long, $RR_{200}$, and $RR_{var}$ in Table 9. LOO-CV performance is
important in this analysis because in a real-time application each 45-second
segment will be classified individually, instead of in a large group. Because
LOO-CV has a comparable performance to the other models we’d run we are more
confident in these features’ ability to represent/classify Atrial
Fibrillation.

###  6.2 Exploratory Analysis

|
---|---
a). Dichotomized MIT-BIH Data with Normal-to-Long Transition, $RR_{200}$, and $RR_{var}$ |  b). Plot 1 of 2017 Challenge Data with Normal-to-Long Transition, $RR_{100}$, and $RR_{var}$
|
c). Plot 2 of 2017 Challenge Data with Normal-to-Long Transition, $RR_{100}$, and $RR_{var}$ |  d). Dichotomized 2017 Challenge Data with Normal-to-Long Transition, $RR_{100}$, and $RR_{var}$
Table 10: Exploratory Figures from MIT-BIH Dataset and 2017 Challenge Dataset

With three covariates the data was reduced to a visible dimension, allowing us
to visually analyze the relationship between the heart-beat rhythms and the
extracted features/covariates.

With the MIT-BIH data good separation between Atrial Fibrillation and not-
Atrial Fibrillation occurs with just Normal-to-Long, $RR_{100}$, and
$RR_{var}$ as seen in Figure 10-a. Although, transitioning to the 2017
Challenge data does not give the same clear separation with Normal-to-Long,
$RR_{var}$, and $RR_{200}$ as the only predictor variables.

In Figure 10-b, with all four classifications, the 2017 Challenge data shows
little to no separation of the data.

When zoomed in further in Figure 10-c it is observed the Normal and Other
heart rhythms follow near identical patterns within the data while Atrial
Fibrillation is slightly away from it. This implies that the heart-rhythms
categorized as other by Physionet are not identifiable with RR-Intervals, or
the features we extracted for this analysis.

In Figure 10-d, with Other and Normal considered the same rhythm, there is
more of difference between the rhythms. While the separation is not as
pronounced as with the MIT-BIH database, finding the right amount of Support
Vectors within our Radial SVM may result in good results.

###  6.3 Transfer Learning Using Support Vector Machines

To analyze transfer learning with the SVM model we built, two new models were
developed. They were a balanced model developed with 5,000 AFib samples and
not-AFib samples from the MIT-BIH Dataset, as well as a weighted model
developed with all samples from the MIT-BIH Dataset. These two models were
then tested on the 2017 Challege Data with each reading once again
dichotomized as AFib or not-AFib for consistency with the MIT-BIH data.

The first model was trained with the balanced dataset of 5,000 AFib and 5,000
not-AFib from the MIT-BIH data set. It was then tested on the 2017 Challenge
data set. In our preliminary studies we tested different parameters and found
cost = 10 and $\gamma=\frac{1}{18}$ had the best performance. This model was
compared against SVM with the entire dataset of 17,613 MIT-BIH samples with
uneven amounts of not-AFib and AFib samples. The class weights which were used
for this are given by

$[AFib,\text{{not-AFib}}]=[\frac{1-\Sigma
AFib}{17,613},\frac{1-\Sigma\text{{not-AFib}}}{17,613}],$ (10)

where $\Sigma\text{{not-AFib}}$ and $\Sigma AFib$ are the number of not-AFib
or AFib samples, respectively, within the training set.

|  Accuracy |  Sensitivity |  Specificity
---|---|---|---
Balanced SVM | 84.28% | 92.44% | 53.46%
Weighted SVM | 83.95% | 93.13% | 51.52%
Table 11: SVM Testing Results

As seen in Table 11, the balanced SVM model performed slightly better in
sensitivity than the weighted model. To address this models are trained and
tested on the 2017 Challenge data to filter out any cross database model
problems.

To classify the 2017 Challenge data a Support Vector Machine Pipeline, or
multiple Support Vecor Machines, described below, was required to account for
each classification provided with the data.

  1. 1.

Train SVM model with Noisy classifications against all alternative
classifications

  2. 2.

Remove what is predicted as Noisy from the Testing Set and what is Noisy in
the Training Set

  3. 3.

Train SVM model with AFib classifications against all remaining
classifications

  4. 4.

Remove what is predicted as AFib from the Testing Set and what is Noisy in the
Training Set

  5. 5.

Train SVM model with Other classifications vs all remaining classifications

  6. 6.

Remove what is predicted as Other from the Testing Set

  7. 7.

Classify left-over data as Normal

With this “pipeline” we performed 5-fold Cross-Validation on the 2017
Challenge Dataset.

|  Accuracy |  Specificity |  Sensitivity
---|---|---|---
Noise | 99.479% | 99.526% | 25.000%
AFib | 92.115% | 94.213% | 58.964%
Other | 75.177% | 78.885% | 61.061%
Table 12: Results of 5-Fold Cross Validation of Pipeline SVM Model using 2017
Challenge Data

Table 12 shows that using the “pipeline” for multiclass classification did not
have high performance, with similar results to training with the MIT-BIH data
set and testing with the 2017 Challenge data set. While the Specificities are
moderate and the Accuracies are not extremely low, the Sensitivities can be
improved for detection of a life threatening disease such as Atrial
Fibrillation. While this approach does have potential for Atrial Fibrillation
diagnosis, the use of all four classifications may be causing inaccuracies in
the SVM models.

Based on the plots of the 2017 Challenge Dataset we ran binary classification
SVM on the 2017 Challenge dataset. The results were much higher, supporting
our previous hypothesis that RR-Intervals and features based on RR-Interval
variation are not sufficient for classifying all Other rhythms.

Confusion Matrix |  Acc |  Spec |  Sens
---|---|---|---
|  | N | AF
---|---|---
N | 6,488 | 142
AF | 1,257 | 626
0.836 | 0.838 | 0.815
Table 13: Binary SVM Results of 2017 Challenge Dataset

##  7 Conclusions

When looking at the data in the MIT-BIH Database and 2017 Challenge Dataset we
were given the RR-Intervals and Annotations of the heart-beat Rhythm Types.
Once these were extracted we could identify features to identify patterns to
understand the data. We extracted features of transitions based on the running
mean and classifying the heart-beats as Normal, Short, or Long compared to the
running mean. Doing this we found that in AFib the transitions are much less
binary and tend to converge when transitioning from the Normal heart-beats to
other classifiers. With this understanding of the data we continued to extract
features such as $RR_{var}$ and $RR_{200}$.

With these features extracted we identified Support Vector Machine with a
radial kernel as a promising model to classify AFib. When tested with the MIT-
BIH database this method worked very well and led us to decrease our
covariates to just Normal-To-Long Transition Proportion, $RR_{var}$, and
$RR_{200}$. The plot of the MIT-BIH data further supported this decision
showing clear separation of AFib and not-AFib rhythms. When switching
databases we did run into some problems.

Plotted, the 2017 Challenge dataset did not show any clear separation with
just three covariates and when tested with SVM trained on the MIT-BIH database
it did not perform in a promising manor. To account for any cross-database
inconsistencies we tested and trained an SVM model on the 2017 Challenge data
in a “pipeline” approach to account for all four classifications supplied with
the data. This approach did increase our statistics, but sensitivity was not
raised to a desirable level for our goal.

With binary classification the sensitivity was increased, implying that RR-
Intervals are not sufficient data for all heart-beat rhythms. Although,
features based on RR-Intervals are well suited for Atrial Fibrillation
classification.

##  8 Future Work

Classification of the MIT-BIH data worked very well with just three predictor
variables and a Radial SVM model. Although, this dataset is very oversampled
and highly processed/cleaned. Cross validation with the 2017 Challenge data
resulted in lower performance when four heart-rhythms were classified, but
increased with binary classification. This implies that while RR-Interval-
based features may be suited to classify Atrial Fibrillation, they are not
suited to classify all heart rhythms. For application in Photoplethysmographs
this hypothesis needs to be cross validated with PPG datasets. While cleaning
of the 2017 Challenge ECG data did improve performance, the same processing
techniques may not work with PPG data.

In the future, this research will be continued to look closer into processing
an classification with PPG data, as well as application into a real time model
that can filter out noise and extract features continuously.

## Acknowledgements

This work was supported by the National Science Foundation under DMS Grant
Number 1659288. I would like to thank Dr. Cuixian Chen and Dr. Yishi Wang for
this opportunity, their support in learning the methods necessary for this
research, and pushing me to learn more. Thank you to my colleagues Christian
Thompson, Jericho Lawson, and Drew Johnston for their input and support
through this process.

## References

  * [1] G. D. Clifford, C. Liu, B. Moody, H. L. Li-wei, I. Silva, Q. Li, A. Johnson, and R. G. Mark.  Af classification from a short single lead ecg recording: the physionet/computing in cardiology challenge 2017.  In 2017 Computing in Cardiology (CinC), pages 1–4. IEEE, 2017.
  * [2] T. Edition and G. B. Moody.  Wfdb applications guide.
  * [3] R. M. GB Moody.  A new method for detecting atrial fibrillation using r-r intervals.  Computers in Cardiology, 10:227–230, 1983.
  * [4] A. Ghodrati and S. Marinello.  Statistical analysis of rr interval irregularities for detection of atrial fibrillation.  In 2008 Computers in Cardiology, pages 1057–1060. IEEE, 2008.
  * [5] A. L. Goldberger, L. A. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E. Stanley.  Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals.  Circulation, 101(23):e215–e220, 2000.
  * [6] D. E. Lake and J. R. Moorman.  Accurate estimation of entropy in very short physiological time series: the problem of atrial fibrillation detection in implanted ventricular devices.  American Journal of Physiology-Heart and Circulatory Physiology, 300(1):H319–H325, 2010.
  * [7] B. Larburu Rubio.  Estudio comparativo de algoritmos para la detección de la fibrilación auricular.  2011\.
  * [8] NIHBLI.  Atrial fibrillation.
  * [9] B. Young, D. Brodnick, and R. Spaulding.  A comparative study of a hidden markov model detector for atrial fibrillation.  In Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No. 98TH8468), pages 468–476. IEEE, 1999.