# Some properties defined by relative versions of star-covering properties II

Maddalena Bonanzinga, Davide Giacopello and Fortunato Maesano

_Dedicated to the memory of Mikhail (Misha) Matveev_

###### Abstract

In this paper we consider some recent relative versions of Menger property
called set strongly star Menger and set star Menger properties and the
corresponding Hurewicz-type properties. In particular, using [2], we ”easily”
prove that the set strong star Menger and set strong star Hurewicz properties
are between countable compactness and the property of having countable extent.
Also we show that the extent of a regular set star Menger or a set star
Hurewicz space cannot exceed $\mathfrak{c}$. Moreover, we construct (1) a
consistent example of a set star Menger (set star Hurewicz) space which is not
set strongly star Menger (set strongly star Hurewicz) and show that (2) the
product of a set star Menger (set star Hurewicz) space with a compact space
need not be set star Menger (set star Hurewicz). In particular, (1) and (2)
answer to some questions posed by Kočinac, Konca and Singh in [16] and [19].

Keywords:  Star compact, strongly star compact, star Lindelöf, strongly star
Lindelöf, star Menger, strongly star Menger, star Hurewicz, strongly star
Hurewicz, set properties.

AMS Subject Classification: 54D20

##  1 Introduction

Let ${\mathcal{U}}$ be a cover of a space $X$ and $A$ be a subset of $X$; the
star of $A$ with respect to ${\mathcal{U}}$ is the set
$st(A,{\mathcal{U}})=\bigcup\\{U:U\in{\mathcal{U}}\;\hbox{and}\;U\cap
A\neq\emptyset\\}$. The star of a one-point set $\\{x\\}$ with respect to a
cover ${\mathcal{U}}$ is denoted by $st(x,{\mathcal{U}})$.

Recall that a space $X$ is star compact, briefly SC (strongly star compact,
briefly SSC) if for every open cover $\mathcal{U}$ of the space $X$, there
exists a finite subfamily $\mathcal{V}$ of $\mathcal{U}$ (resp., a finite
subset $F$ of $X$) such that $st(\bigcup{\mathcal{V}},{\mathcal{U}})=X$
(resp., $st(F,{\mathcal{U}})=X$) (see [10] where another terminology is used,
and [7]); $X$ is star Lindelöf, briefly SL (strongly star Lindelöf, briefly
SSL) if for every open cover $\mathcal{U}$ of the space $X$, there exists a
countable subfamily $\mathcal{V}$ of $\mathcal{U}$ (resp., a countable subset
$C$ of $X$) such that $st(\bigcup{\mathcal{V}},{\mathcal{U}})=X$ (resp.,
$st(C,{\mathcal{U}})=X$) (see [11] and [12], where different terminology is
used).

In [15, 16] Kočinac, Konca and Singh introduced the following relative
versions of SC, SSC, SL and SSL properties.

######  Definition 1.1.

[16] A space $X$ is set star compact, briefly set SC (resp., set strongly star
compact, briefly set SSC), if for every nonempty subset $A$ of $X$ and for
every family $\mathcal{U}$ of open sets in X such that
$\overline{A}\subseteq\bigcup{\mathcal{U}}$, there exists a finite subfamily
$\mathcal{V}$ of $\mathcal{U}$ (resp., finite subset $F$ of $\overline{A}$)
such that $st(\bigcup{\mathcal{V}},{\mathcal{U}})\supset A$ (resp.,
$st(F,{\mathcal{U}})\supset A$).111Recently, the properties of Definition 1.1
were studied in [2]. Note that in [2] there is a misprint in the statment of
the definition of ”relatively∗ SSC” that the authors use to describe set SSC
property: in particular, the authors write that the set ”$F$ is a finite
subset of $A$” instead of ”$F$ is a finite subset of $\overline{A}$”.

Replacing ”finite” with ”countable” in Definition 1.1, one obtains the classes
of set star Lindelöf (briefly set SL) and set strongly star Lindelöf (briefly
set SSL) spaces (see [15]).

In the following CC means countably compact.

######  Proposition 1.1.

[2, Proposition 2.2] In the class of Hausdorff spaces SSC, set SSC and CC are
equivalent properties.

We prove the following

######  Proposition 1.2.

In the class of regular spaces set SC and CC are equivalent properties.

Of course, every CC space is set SC. Now, let $X$ a regular set SC space. By
contradiction, assume there exists a closed and discrete subspace
$D=\\{x_{n}:n\in\omega\\}$ of $X$. By regularity, there exists a disjoint
family ${\cal U}=\\{U_{n}:n\in\omega\\}$ of open subsets of $X$ such that
$x_{n}\in U_{n}$, for every $n\in\omega$. Then
$D\subseteq\bigcup{\mathcal{U}}$ but for every finite subfamily $\mathcal{V}$
of $\mathcal{U}$, we have that $D\not\subset
st(\bigcup{\mathcal{V}},{\mathcal{U}})$; a contradiction. For a space $X$,
$e(X)=\sup\\{|C|\,\,:\,\,C\hbox{ is a closed and discrete subset of }X\\}$ and
$c(X)=\sup\\{|{\cal A}|\,\,:\,\,{\cal A}\hbox{ is a cellular family of }X\\}$
are, respectively, the extent and the cellularity of $X$. One says that a
space $X$ has the countable chain condition (briefly ccc) if $c(X)=\omega$.

######  Proposition 1.3.

[2, Proposition 3.1] In the class of $T_{1}$ spaces, set SSL spaces are
exactly spaces having countable extent.

######  Proposition 1.4.

[2, Corollary 3.3] Every ccc space is set SL.

Recall that a space $X$ is Menger, briefly M, if for each sequence $({\cal
U}_{n}:n\in\omega)$ of open covers of $X$ there exists a sequence $({\cal
V}_{n}:n\in\omega)$ such that ${\mathcal{V}}_{n}$, $n\in\omega$, is a finite
subset of ${\mathcal{U}}_{n}$ and
$X=\bigcup_{n\in\omega}\bigcup{\mathcal{V}}_{n}$; $X$ is Hurewicz, briefly H,
if for each sequence $({\cal U}_{n}:n\in\omega)$ of open covers of $X$ there
exists a sequence $({\cal V}_{n}:n\in\omega)$ such that ${\mathcal{V}}_{n}$,
$n\in\omega$, is a finite subset of ${\mathcal{U}}_{n}$ and for every $x\in
X$, $x\in\bigcup{\mathcal{V}}_{n}$ for all but finitely many $n\in\omega$.

In [13, 14, 1] star versions of Menger and Hurewicz properties called star
Menger, strongly star Menger, star Hurewicz and strongly star Hurewicz
properties (Definitions 2.1 and 4.1 below) were introduced and recently in
[16] Kočinac, Konca and Singh considered some relative versions of them
called, respectively, set star Menger, set strongly star Menger, set star
Hurewicz and set strongly star Hurewicz properties.

In this paper we study the previous set properties. In particular, using [2],
we easily prove that set strongly star Menger and set strongly star Hurewicz
properties are between countable compactness and property of having countable
extent. Also we show that the extent of a regular set star Menger or a set
star Hurewicz space cannot exceed $\mathfrak{c}$ and use this result to give a
Tychonoff star Menger (star Hurewicz) space which is not set star Menger (set
star Hurewicz). In fact, the constructed example (Example 2.3) is even star
compact and then it gives a positive answer to the following question.

######  Question 1.1.

[19] Does there exist a Tychonoff star compact space which is not set star
compact?

Moreover, we give a consistent answer (Example 2.5) to the following question.

######  Question 1.2.

[16] Does there exist a Tychonoff set star Menger space which is not set
strongly star Menger?

Further, we answer in the negative (Example 3.1) to the following

######  Question 1.3.

[16] Is the product of a set star Menger space with a compact space a set star
Menger space?

In fact Example 3.1 shows even more: it proves that set star compact and set
star Lindelöf properties are not preserved in the product with compact spaces.
Then, the same example answers in the negative to the following two questions.

######  Question 1.4.

Is the product of a set star Hurewicz space with a compact space a set star
Hurewicz space?

######  Question 1.5.

[19] Is the product of a set star compact space with a compact space a set
star compact space?

Moreover we give partial answers to the following questions.

######  Question 1.6.

[16] Is the product of a set strongly star Menger space with a compact space a
set strongly star Menger space?

######  Question 1.7.

Is the product of a set strongly star Hurewicz space with a compact space a
set strongly star Hurewicz space?

No separation axiom will be assumed a priori. Recall that a family of sets is
almost disjoint if the intersection of any two distinct elements is finite.
Let $\cal A$ be an almost disjoint family of infinite subsets of $\omega$. Put
$\Psi({\mathcal{A}})=\omega\cup{\mathcal{A}}$ and topologize
$\Psi({\mathcal{A}})$ as follows: the points of $\omega$ are isolated and a
basic neighbourhood of a point $a\in\mathcal{A}$ takes the form
$\\{a\\}\cup(a\setminus F)$, where $F$ is a finite set. $\Psi({\mathcal{A}})$
is called Isbell-Mrówka or $\Psi$-space (see [8]). Recall that for
$f,g\in\omega^{\omega}$, $f\leq^{*}g$ means that $f(n)\leq g(n)$ for all but
finitely many $n$ (and $f\leq g$ means that $f(n)\leq g(n)$ for all
$n\in\omega$). A subset $B\subseteq\omega^{\omega}$ is bounded if there is
$g\in\omega^{\omega}$ such that $f\leq^{*}g$ for every $f\in B$.
$D\subseteq\omega^{\omega}$ is cofinal if for each $g\in\omega^{\omega}$ there
is $f\in D$ such that $g\leq^{*}f$. The minimal cardinality of an unbounded
subset of $\omega^{\omega}$ is denoted by ${\mathfrak{b}}$, and the minimal
cardinality of a cofinal subset of $\omega^{\omega}$ is denoted by
${\mathfrak{d}}$. The value of ${\mathfrak{d}}$ does not change if one
considers the relation $\leq$ instead of $\leq^{*}$ [6, Theorem 3.6].

##  2 On set star Menger and set strongly star Menger properties.

In [13], Kočinac introduced the following star versions of Menger property.

######  Definition 2.1.

[13] A space $X$ is

$\bullet$ star Menger (briefly, SM) if for each sequence $({\cal
U}_{n}:n\in\omega)$ of open covers of $X$ there exists a sequence $({\cal
V}_{n}:n\in\omega)$ such that ${\mathcal{V}}_{n}$, $n\in\omega$, is a finite
subset of ${\mathcal{U}}_{n}$ and
$X=\bigcup_{n\in\omega}st(\bigcup{\mathcal{V}}_{n},{\mathcal{U}}_{n})$;

$\bullet$ strongly star Menger (briefly, SSM) if for each sequence $({\cal
U}_{n}:n\in\omega)$ of open covers of $X$ there exists a sequence
$(F_{n}:n\in\omega)$ such that $F_{n}$, $n\in\omega$, is a finite subset of
$X$ and $X=\bigcup_{n\in\omega}st(F_{n},{\mathcal{U}}_{n})$.

The following result gives a characterization of the SSM property in terms of
a relative version of it.

######  Proposition 2.1.

The following are equivalent for a space $X$:

  1. 1.

$X$ is SSM;

  2. 2.

for each nonempty subset $A$ of $X$ and each sequence $({\cal
U}_{n}:n\in\omega)$ of collection of open sets of $X$ such that
$\overline{A}\subset\bigcup{\mathcal{U}}_{n}$ for every $n\in\omega$, there
exists a sequence $(F_{n}:n\in\omega)$ such that $F_{n}$, $n\in\omega$, is a
finite subset of $X$ and
$A\subset\bigcup_{n\in\omega}st(F_{n},{\mathcal{U}}_{n})$.

$2.\Rightarrow 1.$ is obvious. Let $A\subseteq X$ be a nonempty subset and
$({\cal U}_{n}:n\in\omega)$ be a sequence of families of open sets of $X$ such
that $\overline{A}\subseteq\bigcup{\mathcal{U}}_{n}$ for every $n\in\omega$.
Define

${\mathcal{U}}^{{}^{\prime}}_{n}={\mathcal{U}}_{n}\cup\\{X\setminus\overline{A}\\}$

for all $n\in\omega$. Clearly, each ${\mathcal{U}}^{{}^{\prime}}_{n}$ is an
open cover for $X$. Since $X$ is SSM, there is a sequence $(F_{n}:n\in\omega)$
of finite subsets of $X$ such that
$X=\bigcup_{n\in\omega}st(F_{n},{\mathcal{U}}_{n})$. Fix $x\in A$. Then there
exists $n\in\omega$ such that $x\in
st(F_{n},{\mathcal{U}}^{{}^{\prime}}_{n})$. Observe that

$x\in st(F_{n},{\mathcal{U}}^{{}^{\prime}}_{n})\Leftrightarrow
st(x,{\mathcal{U}}^{{}^{\prime}}_{n})\cap F_{n}\neq\emptyset$

We also have that

$st(x,{\mathcal{U}}^{{}^{\prime}}_{n})=\bigcup\\{U\in{\mathcal{U}}^{{}^{\prime}}_{n}:x\in
U\\}=\bigcup\\{U\in{\mathcal{U}}_{n}:x\in U\\}=st(x,{\mathcal{U}}_{n})$

So

$x\in st(F_{n},{\mathcal{U}}^{{}^{\prime}}_{n})\Leftrightarrow
st(x,{\mathcal{U}}^{{}^{\prime}}_{n})\cap F_{n}\neq\emptyset\Leftrightarrow
st(x,{\mathcal{U}}_{n})\cap F_{n}\neq\emptyset\Leftrightarrow x\in
st(F_{n},{\mathcal{U}}_{n})$.

Since $x$ is an arbitrary point of $A$, we have that
$A\subset\bigcup_{n\in\omega}st(F_{n},{\mathcal{U}}_{n})$.

In [16] the following relative version of the SM and SSM properties were
considered.

######  Definition 2.2.

[16] A space $X$ is

$\bullet$ set star Menger (shortly, set SM) if for each nonempty subset $A$ of
$X$ and for each sequence $({\cal U}_{n}:n\in\omega)$ of collection of open
sets of $X$ such that $\overline{A}\subset\bigcup{\mathcal{U}}_{n}$ for every
$n\in\omega$, there exists a sequence $({\cal V}_{n}:n\in\omega)$ such that
${\mathcal{V}}_{n}$, $n\in\omega$, is a finite subset of ${\mathcal{U}}_{n}$
and
$A\subset\bigcup_{n\in\omega}st(\bigcup{\mathcal{V}}_{n},{\mathcal{U}}_{n})$.

$\bullet$ set strongly star Menger (shortly, set SSM) if for each nonempty
subset $A$ of $X$ and for each sequence $({\cal U}_{n}:n\in\omega)$ of
collection of open sets of $X$ such that
$\overline{A}\subset\bigcup{\mathcal{U}}_{n}$ for every $n\in\omega$, there
exists a sequence $(F_{n}:n\in\omega)$ such that $F_{n}$, $n\in\omega$, is a
finite subset of $\overline{A}$ and
$A\subset\bigcup_{n\in\omega}st(F_{n},{\mathcal{U}}_{n})$.

The following result is easy to check.

######  Proposition 2.2.

A space $X$ is set SSM iff every closed subspace of $X$ is SSM.

The previous result is not true for set SM spaces as the following example
shows.

######  Example 2.1.

A set SM space having a closed subspace which is not SM.

Consider the set SM space $X$ of Example 3.1 below and its closed subspace
$A$. Since $A$ is a discrete subspace of uncountable cardinality, it is not
SM. $\hfill\triangle$

Recall that in [21] it is proved that the extent of a $T_{1}$ SSM space can be
arbitrarily big. Also

######  Proposition 2.3.

[18, Corollary 2.2] Every closed and discrete subspace of a regular SSM space
has cardinality less than $\mathfrak{c}$. Hence a SSM space has extent less or
equal to $\mathfrak{c}$.

It is well known that a CC space has countable extent. Since every CC space is
set SSC (see [7, Theorem 2.1.4] and recall that CC property is hereditary with
respect to closed sets) and every set SSL space has countable extent [2,
Proposition 3.1] we have that

$\hbox{CC}\Rightarrow\hbox{set SSH}\Rightarrow\hbox{set
SSM}\Rightarrow\hbox{countable extent}.$

Note that the previous implications can not be reversed. Indeed, of course,
every Hurewicz non countably compact space is a set SSH non countably compact
space: consider, for example, the discrete space $\omega$. For the converse of
the other implications see Examples 4.1 and 2.4.

In [17] it was shown that the extent of a Tychonoff SSL space can be
arbitrarily large (note that in [17] a SSL space is called a space with
countable weak extent). In [18] the space constructed in [17] was used to
prove that the extent of a Tychonoff SM (in fact SC) space can be arbitrarily
large. Moreover in [18] the author shows the following

######  Theorem 2.1.

[18] If $X$ is a regular SM space such that $w(X)=\mathfrak{c}$, then every
closed and discrete subspace of $X$ has cardinality less than $\mathfrak{c}$.
Hence, we have $e(X)\leq\mathfrak{c}$.

Now we show that the extent of a regular set SM space cannot exceed
$\mathfrak{c}$.

######  Theorem 2.2.

If $X$ is a regular set SM space, then every closed and discrete subspace of
$X$ has cardinality less than $\mathfrak{c}$. Hence, we have
$e(X)\leq\mathfrak{c}$.

Fix $Y$ a closed and discrete subspace of $X$ and assume $|Y|=\mathfrak{c}$.
Consider a family $\cal B$ of open subsets of $X$ such that for every $y\in Y$
there exists $B\in{\cal B}$ such that $y\in B$ and $\overline{B}\cap
Y=\\{y\\}$ and suppose that $|\cal B|=\mathfrak{c}$. Denote by $[\cal
B]^{<\omega}$ the family of all finite subsets of $\cal B$, by
$\mathbb{P}=([\cal B]^{<\omega})^{\omega}$ the family of all the sequences of
elements of $[\cal B]^{<\omega}$ and introduce on $\mathbb{P}$ the partial
order ”$\leq$” defined as follows: if $({\cal
B}_{n}^{\prime})_{n\in\omega},({\cal
B}_{n}^{\prime\prime})_{n\in\omega}\in{\mathbb{P}}\hbox{ then }({\cal
B}_{n}^{\prime})_{n\in\omega}\leq({\cal
B}_{n}^{\prime\prime})_{n\in\omega}\hbox{ means }{\cal
B}_{n}^{\prime}\subseteq{\cal B}_{n}^{\prime\prime}\hbox{ for every
}n\in\omega$. Let $\\{({\cal
B}_{\alpha,n})_{n\in\omega}:\alpha<\mathfrak{c}\\}$ be a cofinal family in
$(\mathbb{P},\leq)$. Take $Z=\\{y_{\alpha}:\alpha<\mathfrak{c}\\}$ by choosing
for every $\alpha<\mathfrak{c}$ a point $y_{\alpha}\in
Y\setminus\bigcup_{n\in\omega}\overline{\bigcup{\cal B}_{\alpha,n}}$ and
$y_{\alpha}\not=y_{\beta}$ for $\alpha\not=\beta$. For every
$\alpha<\mathfrak{c}$ let $\\{V_{n}(y_{\alpha}):n\in\omega\\}$ be a sequence
of open neighbourhoods of $y_{\alpha}$ such that $V_{n}(y_{\alpha})\subseteq
B$ for some $B\in{\cal B}$ and every $n\in\omega$ and

$V_{n}(y_{\alpha})\cap\bigcup{\cal B}_{\alpha,n}=\emptyset$ for every
$n\in\omega$. For every $n\in\omega$ put ${\cal
U}_{n}=\\{V_{n}(y_{\alpha}):\alpha<\mathfrak{c}\\}$. Clearly
$Z=\overline{Z}\subseteq\bigcup{\cal U}_{n}$ for every $n\in\omega$. We will
show that the subset $Z$ and the sequence $({\cal U}_{n}:n\in\omega)$ do not
satisfy the set SM property. Let $({\cal V}_{n}:n\in\omega)$ be any sequence
of finite subsets of ${\cal U}_{n}$ for every $n\in\omega$. Let $({\cal
B}_{n}^{\prime}:n\in\omega)\in\mathbb{P}$ such that every member of ${\cal
V}_{n}$ is contained in a member of ${\cal B}_{n}^{\prime}$. Since $\\{({\cal
B}_{\alpha,n})_{n\in\omega}:\alpha<\mathfrak{c}\\}$ is a cofinal family in
$\mathbb{P}$, there exists $\gamma<\mathfrak{c}$ such that ${\cal
B}_{n}^{\prime}\subseteq{\cal B}_{\gamma,n}$ for every $n\in\omega$. Then
$V_{n}(y_{\gamma})\cap\bigcup{\cal V}_{n}\subseteq
V_{n}(y_{\gamma})\cap\bigcup{\cal B}_{\gamma,n}=\emptyset$ for every
$n\in\omega$. Since $V_{n}(y_{\gamma})$ is the only member of ${\cal U}_{n}$
containing $y_{\gamma}$, we have
$y_{\gamma}\not\in\bigcup_{n\in\omega}st(\bigcup{\cal V}_{n},{\cal U}_{n})$.

Example 2.2 below gives a consistent example of a SSM not set SSM space. In
fact, such an example was already described in [16]; here we show that it can
be easily obtained from the next characterization and from the fact that set
SSM spaces have countable extent.

######  Theorem 2.3.

[4] The following are equivalent:

  * (i)

$\Psi({\cal A})$ is SSM

  * (ii)

$|{\cal A}|<\mathfrak{d}$.

######  Example 2.2.

[16] ($\omega_{1}<\mathfrak{d}$) There exists a SSM not set SSM space.

Assume $\omega_{1}<\mathfrak{d}$ and consider $\Psi({\cal A})$ with $|{\cal
A}|=\omega_{1}$. By Theorem 2.3 and since $e(\Psi({\cal A}))>\omega$, we have
that $\Psi({\cal A})$ is a SSM not set SSM space. $\hfill\triangle$

######  Question 2.1.

Does there exist a ZFC example of a SSM not set SSM space?

Using Theorem 2.2 we can give a Tychonoff space distinguishing SM and set SM
properties. In fact, the following example distinguishes SCness and set SCness
too.

######  Example 2.3.

A Tychonoff SC (hence SM) space which is not set SM (hence not set SC).

In [17], for each infinite cardinal $\tau$ the following space $X(\tau)$ was
considered. Let $Z=\\{f_{\alpha}:\alpha<{\mathfrak{\tau}}\\}$ where
$f_{\alpha}$ denotes the points in $2^{\tau}$ with only the $\alpha$th
coordinate equal to 1. Consider the set

$X(\tau)=(2^{\mathfrak{\tau}}\times({\tau}^{+}+1))\setminus((2^{\tau}\setminus
Z)\times\\{{\tau}^{+}\\})$

with the topology inherited from the product topology on
$2^{\tau}\times({\tau}^{+}+1)$. Denote $X_{0}=2^{\tau}\times{\tau}^{+}$ and
$X_{1}=Z\times{\\{\tau}^{+}\\}$. Then $X(\tau)=X_{0}\cup X_{1}$. $X_{1}$ is a
closed and discrete subspace of $X(\tau)$ of cardinality $\tau$. So the extent
of $X(\tau)$ is $\tau$.
In [18] it is proven that the space $X(\mathfrak{c})$ is SC (hence SM). By
Theorem 2.2, $X(\mathfrak{c})$ it is not set SM. $\hfill\triangle$

Recall the following result:

######  Proposition 2.4.

[18] Every SL (SSL) space of cardinality less than $\mathfrak{d}$ is SM (SSM).

Now we prove that the set versions of the previous proposition holds.

######  Proposition 2.5.

Every set SL space of cardinality less than $\mathfrak{d}$ is set SM.

Let $X$ be a set SL space of cardinality less than $\mathfrak{d}$. Let
$A\subseteq X$ and $({\cal U}_{n}\,\,:\,\,n\in\omega)$ be a sequence of
families of open sets of $X$ such that $\overline{A}\subseteq\bigcup{\cal
U}_{n}$ for every $n\in\omega$. For every $n\in\omega$ there is a countable
subfamily ${\cal V}_{n}=\\{V_{n,m}\,\,:\,\,m\in\omega\\}$ of ${\cal U}_{n}$
such that $A\subseteq st(\bigcup{\cal V}_{n},{\cal U}_{n})$. For every $x\in
A$ we choose a function $f_{x}\in\omega^{\omega}$ such that $st(x,{\cal
U}_{n})\cap V_{n,f_{x}(n)}\not=\emptyset$ for all $n\in\omega$. Since
$\\{f_{x}\,\,:\,\,x\in A\\}$ is not a cofinal family in
$(\omega^{\omega},\leq)$, there are some $g\in\omega^{\omega}$ and
$n_{x}\in\omega$ for $x\in A$ such that $f_{x}(n_{x})<g(n_{x})$. Let ${\cal
W}_{n}=\\{V_{n,j}\,\,:\,\,j\leq g(n)\\}$. Then
$A\subseteq\bigcup_{n\in\omega}st(\bigcup{\cal W}_{n},{\cal U}_{n})$.

In a similar way we can prove the following

######  Proposition 2.6.

Every set SSL space of cardinality less than $\mathfrak{d}$ is set SSM.

Then, by Proposition 1.3 we obtain

######  Corollary 2.1.

For every $T_{1}$ space $X$ of cardinality less than $\mathfrak{d}$, the
following are equivalent:

  1. 1.

$X$ is set SSM

  2. 2.

$e(X)=\omega$.

######  Example 2.4.

There is a Tychonoff space of cardinality $\mathfrak{d}$ having countable
extent which is not set SSM.

Let $\mathbb{P}$ the space of irrationals. Take any non-Menger subspace
$X\subset\mathbb{P}$ of cardinality $\mathfrak{d}$ (for instance, consider the
Baire space $\omega^{\omega}$ which is homeomorphic to $\mathbb{P}$ and take a
cofinal subset of cardinality $\mathfrak{d}$. It is well known that any
cofinal subset of $\omega^{\omega}$ is not Menger). Of course, $X$ is a
paracompact space having countable extent. Since in the class of paracompact
Hausdorff spaces we have that M $\Leftrightarrow$ SM (see [13]), we have that
$X$ is not set SSM. $\hfill\triangle$

By Corollary 2.1 and Example 2.4 we have:

######  Corollary 2.2.

The following statements are equivalent:

  1. 1.

$\omega_{1}<\mathfrak{d}$;

  2. 2.

every $T_{1}$ space of cardinality $\omega_{1}$ having countable extent is set
SSM.

Recall the following result.

######  Theorem 2.4.

[18] The following statements are equivalent for regular spaces.

  1. 1.

$\omega_{1}={\mathfrak{d}}$;

  2. 2.

if $X$ is a SSM space, then $e(X)\leq\omega$.

Now we prove

######  Theorem 2.5.

The following statements are equivalent for regular spaces.

  1. 1.

$\omega_{1}={\mathfrak{d}}$;

  2. 2.

if $X$ is a SSM space, then $e(X)\leq\omega$;

  3. 3.

for spaces of cardinality less than ${\mathfrak{d}}$, set SSM and SSM are
equivalent properties.

  4. 4.

for spaces of cardinality less than ${\mathfrak{d}}$, set SSL and SSL are
equivalent properties.

  5. 5.

every closed subspace of a SSM space $X$ such that $|X|<{\mathfrak{d}}$ is
SSM.

$1.\Leftrightarrow 2.$ holds by Theorem 2.4. Now we prove $2.\Rightarrow 3.$.
Let $X$ be a space of cardinality less than ${\mathfrak{d}}$. By 2. and
Corollary 2.1, we have that $X$ is SSM iff $X$ is set SSM. Now we prove
$3.\Rightarrow 1.$. Assume $\omega_{1}<{\mathfrak{d}}$. Consider a space
$\Psi({\cal A})$, with $|{\cal A}|=\omega_{1}$. By Theorem 2.3, $\Psi({\cal
A})$ is SSM, and since $e(\Psi({\cal A}))>\omega$, $\Psi({\cal A})$ is not set
SSM. $3.\Leftrightarrow 4.$ is obvious. $3.\Leftrightarrow 5.$ follows from
Proposition 2.2.

Of course, countable spaces are Menger, then set SSM and SSM.

######  Corollary 2.3.

For regular spaces $X$ such that $\omega<|X|<{\mathfrak{d}}$, SSM and set SSM
are not equivalent properties.

######  Corollary 2.4.

Uncountable regular spaces in which SSM and set SSM are equivalent properties
have cardinality $\geq{\mathfrak{d}}$.

In [16, Example 5] Kočinac, Konca and Singh constructed a $T_{1}$ set SM space
which is not set SSM and posed the following question.

######  Question 2.2.

[16] Does there exist a Tychonoff set SM space which is not set SSM?

Using Proposition 2.5 we can give a consistent answer to Question 2.2.

######  Example 2.5.

($\omega_{1}<\mathfrak{d}$) A Tychonoff set SM space which is not set SSM.

Assume $\omega_{1}<\mathfrak{d}$ and consider $\Psi(\mathcal{A})$ with
$|{\mathcal{A}}|=\omega_{1}$. Since $\Psi(\mathcal{A})$ is separable, it is
set SL hence, by Proposition 2.5, it is set SM. Since
$e(\Psi(\mathcal{A}))>\omega$, $\Psi(\mathcal{A})$ is not set SSM.
$\hfill\triangle$

##  3 On the product of set SM and set SSM with compact spaces.

Recall that the product of a SC (SSC) space with a compact space is SC (SSC)
([9], [7]); further the product of a SL space with a compact space is SL [7]
while the product of a SSL space with a compact space need not be SSL [7,
Example 3.3.4]. In [13] Kočinac proved that the product of a SM space with a
compact space is SM. Using [3, Lemma 2.3], Matveev noted that assuming
$\omega_{1}<\mathfrak{d}$, if $X=\Psi({\cal A})$ with
$|{\mathcal{A}}|=\omega_{1}$ and $Y$ is a compact space such that
$c(Y)>\omega$, then the product $X\times Y$ is not SSL, hence not SSM;
therefore he gave a consistent example of a not SSM space which is the product
of a SSM space and a compact space. Then, it is natural to consider the
following questions.

######  Question 3.1.

[16] Is the product of a set SSM space with a compact space a set SSM space?

######  Question 3.2.

[16] Is the product of a set SM space with a compact space a set SM space?

In the following we give a partial answer to Question 3.1 and a negative
answer to Question 3.2. Note that we also show that set SSL property is
preserved in the $T_{1}$ product with compact spaces and that set SC and set
SL properties are not preserved in the product with compact spaces. (For
completness, we note that, by Proposition 1.1, set SSC property is preserved
in the Hausdorff product with compact spaces).

The following fact can be easly checked (we give the proof for sake of
completeness). Recall that a map is perfect if it is continuous, closed, onto
and each fiber is compact.

######  Proposition 3.1.

If $f:X\to Y$ is a perfect map and $A$ is an uncountable closed and discrete
subspace of $X$, then $f(A)$ is an uncountable closed and discrete subspace of
$Y$.

Let $f$ and $A$ as in the hypothesis. Clearly $f(A)$ is closed in $Y$. Note
that, for every $y\in f(A)$, $f^{-1}(y)\cap A$ is a closed subset of the
compact subspace $f^{-1}(y)$ and then, since $A$ is discrete, it is finite.
Then $A$ is countable, otherwise $f(A)$ is countable. Now, fix $y\in f(A)$ and
say $f^{-1}(y)\cap A=\\{x_{1},...,x_{n}\\}$. For every $i=1,...,n$ fix an open
subset $U_{i}$ of $X$ such that $A\cap U_{i}=\\{x_{i}\\}$ and put
$U=\bigcup_{i=1}^{n}U_{i}$. Since $A\setminus U$ is a closed subset of $X$, we
have that $f(A\setminus U)=f(A)\setminus\\{y\\}$ is a closed subset of Y, and
then $\\{y\\}$ is open in $f(A)$ with the topology inherited from $Y$.

By the previous proposition, we obtain the following result.

######  Corollary 3.1.

The product of a space having countable extent with a compact space has
countable extent.

Let $X$ be a space with countable extent and $Y$ be a compact space. The
projection from $X\times Y$ onto $X$ is a perfect map. Then, by Proposition
3.1, $e(X\times Y)=\omega$.

By Proposition 1.3, the previous result can be restated as follows.

######  Proposition 3.2.

The $T_{1}$ product of a set SSL space with a compact space is set SSL.

######  Corollary 3.2.

The product of a set SSM space with a compact space has countable extent.

Then, by Corollary 2.1 we have

######  Corollary 3.3.

The $T_{1}$ product of cardinality less than $\mathfrak{d}$ of a set SSM space
with a compact space is set SSM.

Recall the following proposition.

######  Proposition 3.3.

[2, Proposition 3.4] Let $X$ be a space. If there exist a closed and discrete
subspace $D$ of $X$ having uncountable cardinality and a disjoint family
${\mathcal{U}}=\\{O_{a}:a\in D\\}$ of open neighbourhoods of points $a\in D$,
then $X$ is not set SL.

Now we prove the following useful result.

######  Proposition 3.4.

If $e(X)>\omega$ and $c(Y)>\omega$, where $Y$ is $T_{1}$, then $X\times Y$ is
not set SL.

Let $S=\\{s_{\alpha}:\alpha<\omega_{1}\\}$ be a closed and discrete subset of
$X$, ${\mathcal{O}}=\\{O_{\alpha}:\alpha<\omega_{1}\\}$ be a pairwise disjoint
family of nonempty open subsets of $Y$. For every $\alpha<\omega_{1}$, fix
$t_{\alpha}\in O_{\alpha}$. Put
$A=\\{(s_{\alpha},t_{\alpha}):\alpha<\omega_{1}\\}$. It is obvious that $A$ is
an uncountable discrete subspace of $X\times Y$. Now we prove that $A$ is
closed. For every $\alpha<\omega_{1}$ there exists an open set, say
$N_{\alpha}$, such that $N_{\alpha}\cap S=\\{s_{\alpha}\\}$. Then $(X\times
Y)\setminus A=((X\setminus S)\times
Y)\cup\bigcup_{\alpha<\omega_{1}}(N_{\alpha}\times(Y\setminus\\{t_{\alpha}\\}))$.
Then, by Proposition 3.3, $X\times Y$ is not set SL.

######  Example 3.1.

There exists a set SC (hence set SL and set SM) space $X$ and a compact space
$Y$ with $c(Y)>\omega$ such that $X\times Y$ is not set SL (hence neither set
SM nor set SC).

Consider the set $X=\omega_{1}\cup A$, where
$A=\\{a_{\alpha}:\alpha\in\omega_{1}\\}$ is a set of cardinality $\omega_{1}$,
topologized as follows: $\omega_{1}$ has the usual order topology and is an
open subspace of $X$; a basic neighborhood of a point $a_{\alpha}\in A$ takes
the form

$O_{\beta}(a_{\alpha})=\\{a_{\alpha}\\}\cup(\beta,\omega_{1}),\hbox{ where
}\beta<\omega_{1}.$

In [2] it was proved that $X$ is set SC, hence $X$ is set SM. We have that
$e(X)>\omega$. If $Y$ is any compact space with $c(Y)>\omega$, by Proposition
3.4, $X\times Y$ is not set SL. $\hfill\triangle$

##  4 On set star Hurewicz and set strongly star Hurewicz properties.

Recall the following definitions.

######  Definition 4.1.

[13, 1] A space $X$ is

$\bullet$ star Hurewicz (briefly, SH) if for each sequence $({\cal
U}_{n}:n\in\mathbb{N})$ of open covers of $X$ there exists a sequence $({\cal
V}_{n}:n\in\mathbb{N})$ such that ${\mathcal{V}}_{n}$, $n\in\omega$, is a
finite subset of ${\mathcal{U}}_{n}$ and $\forall x\in X$, $x\in
st(\bigcup{\mathcal{V}}_{n},{\mathcal{U}}_{n})$ for all but finitely many
$n\in\omega$;

$\bullet$ strongly star Hurewicz (briefly, SSH) if for each sequence $({\cal
U}_{n}:n\in\mathbb{N})$ of open covers of $X$ there exists a sequence
$(F_{n}:n\in\mathbb{N})$ such that $F_{n}$, $n\in\omega$, is a finite subset
of $X$ and $\forall x\in X$, $x\in st(F_{n},{\mathcal{U}}_{n})$ for all but
finitely many $n\in\omega$.

The following result is a characterization of SSH property in terms of a
relative version of it. The proof is similar to the proof of Proposition 2.1.

######  Proposition 4.1.

The following are equivalent for a space $X$:

  1. 1.

$X$ is SSH;

  2. 2.

for each nonempty subset $A$ of $X$ and for each sequence $({\cal
U}_{n}:n\in\mathbb{N})$ of collection of open sets of $X$ such that
$\overline{A}\subset\bigcup{\mathcal{U}}_{n}$ for every $n\in\omega$, there
exists a sequence $(F_{n}:n\in\mathbb{N})$ such that $F_{n}$, $n\in\omega$, is
a finite subset of $X$ and $\forall x\in A$, $x\in
st(F_{n},{\mathcal{U}}_{n})$ for all but finitely many $n\in\omega$.

######  Definition 4.2.

[16] A space $X$ is

  * •

set star Hurewicz (briefly, set SH) if for each nonempty subset $A\subseteq X$
and for each sequence $({\mathcal{U}}_{n}:n\in\omega)$ of collection of open
sets of $X$ such that $\overline{A}\subseteq\bigcup{\mathcal{U}}_{n}$ for
every $n\in\omega$, there exists a sequence $({\cal V}_{n}:n\in\mathbb{N})$
such that ${\mathcal{V}}_{n}$, $n\in\omega$, is a finite subset of
${\mathcal{U}}_{n}$ and $\forall x\in A$, $x\in
st(\bigcup{\mathcal{V}}_{n},{\mathcal{U}}_{n})$ for all but finitely many
$n\in\omega$.

  * •

set strongly star Hurewicz (briefly, set SSH) if for each nonempty subset
$A\subseteq X$ and for each sequence $({\mathcal{U}}_{n}:n\in\omega)$ of
collection of open sets of $X$ such that
$\overline{A}\subseteq\bigcup{\mathcal{U}}_{n}$ for every $n\in\omega$, there
exists a sequence $(F_{n}:n\in\omega)$ such that $F_{n}$, $n\in\omega$, is a
finite subset of $\overline{A}$ and $\forall x\in A$, $x\in
st(F_{n},{\mathcal{U}}_{n})$ for all but finitely many $n\in\omega$.

######  Example 4.1.

($\mathfrak{b}<\mathfrak{d}$) There is a Tychonoff set SSM space which is not
set SSH.

Consider an unbounded subset $X$ of the Baire space $\omega^{\omega}$ of
cardinality $\mathfrak{b}$. Then $X$ is not Hurewicz and, by Corollary 2.1,
$X$ is set SSM. Since $X$ is a paracompact space and in the class of
paracompact Hausdorff spaces we have that H $\Leftrightarrow$ SH (see [1]), we
have that $X$ is not set SSH. $\hfill\triangle$

Recall the following characterization of SSH spaces.

######  Theorem 4.1.

[4] The following properties are equivalent:

  * (i)

$\Psi({\cal A})$ is SSH

  * (ii)

$|{\cal A}|<\mathfrak{b}$.

Then, we can ”easily” give the following result (the same example was given in
[20] using a longer proof).

######  Example 4.2.

($\omega_{1}<\mathfrak{b}$) There exists a SSH not set SSH space.

Assume $\omega_{1}<\mathfrak{b}$ and consider $\Psi({\cal A})$ with $|{\cal
A}|=\omega_{1}$. Then, by Theorem 4.1 and since $e(\Psi({\cal A}))>\omega$, we
have that $\Psi({\cal A})$ is SSH not a set SSH space. $\hfill\triangle$

######  Question 4.1.

Does there exist a ZFC example of a SSH not set SSH space?

By Theorem 2.2 we can give the following

######  Theorem 4.2.

If $X$ is a regular set SH space , then every closed and discrete subspace of
$X$ has cardinality less than $\mathfrak{c}$. Hence, we have
$e(X)\leq\mathfrak{c}$.

In [20, Esempio 2.4] it is given a Hausdorff SH space which is not set SH. Now
we can provide the following

######  Example 4.3.

A Tychonoff SC (hence SH) space which is not set SH.

Consider the space $X(\mathfrak{c})$ of Example 2.3. $X(\mathfrak{c})$ is SC
(hence SH) and, by Theorem 4.2, it is not set SH. $\hfill\triangle$

Recall the following

######  Proposition 4.2.

[5, Corollary 3.10] Every SL (SSL) space of cardinality less than
$\mathfrak{b}$ is SH (SSH).

In analogy to Proposition 2.5 and Proposition 2.6, we can prove the following

######  Proposition 4.3.

Every set SL (set SSL) space of cardinality less than $\mathfrak{b}$ is set SH
(set SSH).

Let $X$ be a set SL space of cardinality less than $\mathfrak{b}$ (the proof
is similar if $X$ is set SSL). Let $A\subseteq X$ and $({\cal
U}_{n}\,\,:\,\,n\in\omega)$ be a sequence of families of open sets of $X$ such
that $\overline{A}\subseteq\bigcup{\cal U}_{n}$ for every $n\in\omega$. For
every $n\in\omega$ there is a countable subfamily ${\cal
V}_{n}=\\{V_{n,m}\,\,:\,\,m\in\omega\\}$ of ${\cal U}_{n}$ such that
$A\subseteq st(\bigcup{\cal V}_{n},{\cal U}_{n})$. For every $x\in A$ we
choose a function $f_{x}\in\omega^{\omega}$ such that $st(x,{\cal U}_{n})\cap
V_{n,f_{x}(n)}\not=\emptyset$ for all $n\in\omega$. Since
$\\{f_{x}\,\,:\,\,x\in A\\}$ is a bounded family in
$(\omega^{\omega},\leq^{*})$, there exists $g\in\omega^{\omega}$ such that for
every $x\in A$ we have that $f_{x}(n)\leq g(n)$ for every but finitely many
$n\in\omega$. Let ${\cal W}_{n}=\\{V_{n,j}\,\,:\,\,j\leq g(n)\\}$. Then for
every $x\in A$ we have that $x\in st(\bigcup{\cal W}_{n},{\cal U}_{n})$ for
all but finitely many $n\in\omega$.

Then, by Proposition 1.3, we have

######  Corollary 4.1.

For every $T_{1}$ space $X$ of cardinality less than $\mathfrak{b}$, the
following are equivalent:

  1. 1.

$X$ is set SSH

  2. 2.

$e(X)=\omega$.

By Corollary 2.1 and Corollary 4.1 we have the following

######  Corollary 4.2.

For spaces $X$ such that $|X|<{\mathfrak{b}}$, the following are equivalent:

  1. 1.

$X$ is set SSM

  2. 2.

$X$ is set SSH

  3. 3.

$e(X)=\omega$.

In [20] the authors give a $T_{1}$ set SH space which is not set SSH. Now we
provide the following

######  Example 4.4.

($\omega_{1}<\mathfrak{b}$) A Tychonoff set SH space which is not set SSH.

Assume $\omega_{1}<\mathfrak{b}$ and consider $\Psi(\mathcal{A})$ with
$|{\mathcal{A}}|=\omega_{1}$. Since $\Psi(\mathcal{A})$ is separable, it is
set SL hence, by Proposition 4.3, it is set SH. Since
$e(\Psi(\mathcal{A}))>\omega$, $\Psi(\mathcal{A})$ is not set SSH.
$\hfill\triangle$

Using Example 3.1 we can show that

######  Proposition 4.4.

Set SH property is not preserved in the product with compact spaces.

By Corollary 3.1 we have that

######  Proposition 4.5.

The product of a set SSH space with a compact space has countable extent.

Then, by Corollary 4.2 we obtain

######  Proposition 4.6.

The $T_{1}$ product of cardinality less than $\mathfrak{b}$ of a set SSH space
with a compact space is set SSH.

The following question is open.

######  Question 4.2.

Is the product of a set SSH space with a compact space a set SSH space?

We give the following useful diagram.

Lindelöf

$\boxed{\begin{array}[]{c}\text{countable extent}\\\
\Updownarrow\text{{\scriptsize$T_{1}$}}\\\ \text{set SSL}\\\ \end{array}}$

SSL

set SL

SL

ccc

SMset SM

SSMset SSMM $\boxed{\begin{array}[]{c}\\\ \text{set SSC}\\\ \\\ \\\ \text{ CC
$\Longleftrightarrow$ SSC }\\\ \\\ \end{array}}$

$T_{2}$set SCregularSCset SHSHSSHHset SSH

Acknowledgements. The authors express gratitude to Masami Sakai for useful
suggestions.

## References

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∎

e1e-mail<EMAIL_ADDRESS>e5e-mail<EMAIL_ADDRESS>e4e-mail:
<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>
11institutetext: Department of Physics, National Defence Academy, Khadakwasla,
Pune-411023, India. 22institutetext: Department of Mathematical and Physical
Sciences, College of Arts and Science, University of Nizwa, Sultanate of Oman.
33institutetext: Department of Physics, Jadavpur University, Kolkata 700032,
India. 44institutetext: Department of Mathematics, Jadavpur University,
Kolkata 700032, India.

# Quark stars in 4-dimensional Einstein-Gauss-Bonnet gravity

Ksh. Newton Singhe1,addr1  S. K. Maurya e5,addr2  Abhisek Duttae4,addr3
Farook Rahamane2,addr4  Somi Aktare3,addr4

(Received: date / Accepted: date)

###### Abstract

The article explores the first exact solution in a four-dimensional EGB-
gravity with an anisotropic matter source. The solution has been found by two
assumptions, a metric potential and the MIT-bag equation of state. Further,
the solution has been tested with several physical constraints. Finally, using
the boundary condition we have also estimated the range of bag constant
$\mathcal{B}$ by varying the mass of the structure and the Gauss-Bonnet
coupling constant $\alpha$. This allows us to check the physical viability of
the solution by comparing it with the existing accepted range of bag constant
$\mathcal{B}$.

###### Keywords:

4D EGB gravity Quark star Anisotropy Stability

††journal: Eur. Phys. J. C

##  1 Introduction

A well established fact among the scientific society is the finest methodology
to explain phenomena occurring in 4-dimensional spacetime is the Einstein
General Relativity which is pertinently governed by the Einstein Hilbert
action ($S_{EH}$) and nevertheless the cosmological constant ($\Lambda_{0}$),
can be represented as -

$S_{EH}=\int d^{D}x\sqrt{-g}\left[\frac{M_{P}^{2}R}{2}-\Lambda_{0}\right]$

Here $M_{P}$ represents planck’s mass and $D=4$ i.e. for $4-D$ spacetime. The
planck’s mass is used in correspondence to express to the gravitational
coupling strength. In the realm of higher dimensional spacetime i.e. $D>4$,
ghosh2021phase  performing quadratic curvature corrections to the Einstein
Hilbert action leads to the Einstein$-$Gauss$-$ Bonnet (EGB) theory.
Lovelock’s Theorems lovelock1971einstein ; lovelock1972four  into higher
dimensions are fundamentally generalisation of Einstein’s Equation in higher
dimensions. The main advantage in Lovelock’s theorem is the equations of
motions are second order variants of the metric which makes it much more
mathematically lucid with respect to the other gravity theories that are 4th
or higher derivatives of the metric. It is an intriguing matter that the
lanczos1938remarkable  Einstein Gauss Bonnet gravity can be classified as a
special case of Lovelock’s theorem, as the quadratic curvature corrections
obtained in this method is through variation of the Gauss - Bonnet term.
Zwiebach zwiebach1985curvature  provided us that Gauss-Bonnet term is the lone
pertinent method to perform higher derivative combination for the leading
correction to Einstein theory, in the low energy effective string theory. The
EGB theory basically amalgamates a single term to Einstein-Hilbert action
which is quadratic in Riemann tensor which naturally serves as the action of
Low energy heterotic string theory wiltshire1986spherically ;
wheeler1986symmetric ; boulware1985string  . The low energy effective string
theory is quite a promising theory as it appears to be a prominent theory of
gravity in higher dimension as it complies higher order curvature terms in the
action. This theory is considered be a contender to remove the singularity
problem in Einstein’s general relativity.

Recently, Glavan and Lin glavan2020einstein  introduced a general covariant
modified version of the theory which sorts out multiple non-trivial problems
occurring in $4-D$ spacetime and higher dimensions. The Gauss-Bonnet term
($\alpha$) in $4-D$ spacetime is nothing but Euler Density and hence a total
derivative and does not effect the gravitational dynamics. But on higher
dimensions, $D>4$, it contains an exterior result of the total derivative
along with a $(D-4)$ form. In the higher dimension scenario, Mardones et al.
mardones1991lovelock  and Torii et al. torii2008n+  expressed that the
contribution of the Gauss-Bonnet term is proportional to a vanishing factor of
$(D-4)$ . Glavan and Lin glavan2020einstein  rescaled the coupling factor i.e.
$\alpha$ to $\frac{\alpha}{(D-4)}$. This rescaling is very much congruous to
general relativity as it maintains equal degrees of freedom in all dimensions
and also rules out the Ostrogradsky instability woodard2015theorem . The
regularization of the Gauss-Bonnet term at $D=4$ providing non-trivial
contribution in dynamics was first shown by Tomozawa tomozawa2011quantum .
This modified gravity theory has versatile applications and hence it appeared
to be quite intriguing to researchers in the field of Astrophysics and
Cosmology. Glavan and Lin glavan2020einstein  themselves applied the theory to
the case of static spherically symmetric vacuum black hole which fetches out
quite a different results from what was obtained for Schwarzchild Black Holes
and is open to debate in terms of acceptability. Gurses et. al gurses2020there
showed that, Einstein Gauss Bonnet gravity does not provides any intrinsic 4D
solution in terms of metric, though the viability of the solutions is
appropriate in $D>4$ scenarios. Further extension of this theory was applied
to evaluate the thermodyanmics of Rotating and non-rotating black holes and
calculate parameters like mass, temperature, entropy etc using EGB gravity.
ghosh2020generating ; konoplya2020stability ; kumar2020hayward ;
kumar2020bardeen ; zhang2020superradiance . Along with that, this theory is
successfully applied for the cases of Black gravitational Lensing of both
strong and weak types ghosh2020generating , thermodynamics of EGB Anti de
Sitter Black Holes mansoori2021thermodynamic , evaluating the Quasi Normal
Modes mishra2020quasinormal , Geodesic motions and for spinning test particles
zhang2020spinning , Hawking radiation zhang2020greybody  and even in wormhole
solutions kimet2020wormholes . Apart from the Black Hole problems, EGB find
its application on case of Neutron Stars. The core of the neutron stars is
still an area not much explored yet. Recent inventions of Neutron stars of
2$M_{\odot}$ pushes the equation of state of matter inside the stars to
farthest extent but the inner sanctum remains a mystery. The determination of
inner core matter is quite non-trivial as the composition varies with the
nature of interaction and quark matter, hyperon matter, Bose Einstein
condensate strange mesons are all considered to be the core matter or might be
present in fraction. But the the invention of 2$M_{\odot}$ Neutron stars like
PSR J1614-2230 [$1.97\pm 0.04M_{\odot}$] zhang2020gw190814  and J0348+0432
[$2.01\pm 0.04M_{\odot}$] antoniadis2013massive  brought criterion on the
nature of the core matter. According to Quantum Chromodynamics [QCD], it
supports the conversion of hadronic matter into deconfined quarks inside the
neutron stars. This invention makes the Bodmer-Witten hypothesis quite
important to the context as it already predicted for a quark-matter component
in the cosmic rays detected from neutron stars bodmer1971collapsed ;
witten1984cosmic . This basically strengthens the presence of Quark Stars.

The MIT bag model is the most common methodology to analyse the quark starts.
The MIT bag model considers the quarks as free particles strongly confined
inside hadrons as any presence of free quark is yet to be detected in particle
physics. The quarks are considered to be free to move inside the hadron and
are treated as free fermi gas. The confinement of quarks i.e. the bag model
for the quarks can have arbitrary shapes only function of bag radius
$\mathcal{R}(\theta,\phi)$ and a normal vector $n(\theta,\phi)$ out of which
spherical shape is the most common natural outcomes. The parameters of the bag
radius ($\mathcal{R}$) is determined by the constraint that the pressure on
each quark is constant inside the bag. This theory has certain limitations as
it’s violates chiral symmetry and Equation of State (EoS) obtained here has no
pertinent reliability on discussing systems related to quark interactions. To
modify the shortcomings, Color flavour locked (CFL) matter has been proposed
by few authors banerjee2021color ; alford1999color  which is considered to be
the ground state of QCD at extreme large densities, though the phase of the
matter is not properly deterministic in this case. Asbell et. al
asbell2017oscillation  proposed a 2-component EoS of state for strange quark
stars which are having mass in the range of $2M_{\odot}$ by studying the non-
radial oscillations emitted out from this sources. This model inspired the EoS
for interacting quarks for quark stars in Einstein-Gauss-Bonnet gravity
banerjee2021quark . The derived EoS was based on simple thermodynamic bag
model which considered it as homogeneous and unpaired charge neutral 3-flavor
interacting quark matter in case of strange quarks.

In this paper, we will try to evaluate exact solution in $4D$ EGB-gravity for
the first time that represents quark star where the internal pressure and
energy density are linked by the famous MIT bag model EoS. In Sect. 2 we will
explore some aspect of $4D$ EGB gravity and its field equations for
anisotropic stress tensor, Sect. 3 deals with physics of QCD and MIT-bag model
for quark matters, a new solution is explored in Sect. 4. The matching
conditions of the solution are provided in Sect. 5 and the Sect. 6 contains
the physical properties of the solution for quark star. At last, finally the
results and discussions are provided in Sect. 7.

##  2 The field equations

The Lagrangian in $n-$dimensional Einstein-Gauss-Bonnet (EGB) gravity is given
by

$\mathcal{L}_{EGB}=\mathcal{R}+{\alpha\over
n-4}~{}\mathcal{L}_{GB}+\mathcal{L}_{m},$ (1)

and corresponding action can be written as

$\displaystyle\mathcal{S}_{EGB}={1\over 16\pi}\int
d^{n}x\sqrt{-g}\Big{(}\mathcal{R}+{\alpha\over
n-4}~{}\mathcal{L}_{GB}\Big{)}+\mathcal{S}_{m}.$ (2)

Here $\mathcal{R}$ is the Ricci scalar, $\alpha$ the Gauss-Bonnet coupling
parameter, $\mathcal{S}_{m}$ is action contributed from the matter
distribution and the Gauss-Bonnet Lagrangian is given by

$\displaystyle\mathcal{L}_{GB}=\mathcal{R}^{2}-4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}+\mathcal{R}_{\mu\nu\kappa\lambda}\mathcal{R}^{\mu\nu\kappa\lambda}.$
(3)

The variation of the action in (2) with respect to the metric tensor gives

$\displaystyle\mathcal{G}_{\mu\nu}+{\alpha\over
n-4}~{}\mathcal{H}_{\mu\nu}=8\pi\mathcal{T}_{\mu\nu}.$ (4)

Here the Einstein tensor, the Lancoz tensor and the stress tensor are
respectively given as

$\displaystyle\mathcal{G}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathcal{R}_{\mu\nu}-{1\over 2}~{}\mathcal{R}~{}g_{\mu\nu}~{},$
(5) $\displaystyle\mathcal{H}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\Big{(}\mathcal{R}\mathcal{R}_{\mu\nu}-2\mathcal{R}_{\mu\lambda}\mathcal{R}^{\lambda}_{\nu}-2\mathcal{R}_{\mu\lambda\nu\rho}\mathcal{R}^{\lambda\rho}-$
(6)
$\displaystyle\mathcal{R}_{\mu\alpha\beta\gamma}\mathcal{R}^{\alpha\beta\gamma}_{\nu}\Big{)}-{1\over
2}~{}g_{\mu\nu}\mathcal{L}_{GB},$ $\displaystyle\mathcal{T}_{\mu\nu}$
$\displaystyle=$
$\displaystyle-{2\over\sqrt{-g}}~{}{\delta(\sqrt{-g}~{}\mathcal{S}_{m})\over\delta
g^{\mu\nu}}.$ (7)

The field equation (4) cannot give a meaningful field equation at $n=4$.
However, Glavan & Lin D. Glavan and Lin glavan2020einstein  shown that by a
re-scaling $\alpha\rightarrow\alpha/(n-4)$ and Ghosh & Maharaj ghm  by
considering spacetimes of curvature scale $\mathcal{K}$ which are maximally
symmetric determined the variation of the Gauss-Bonnet contribution as

$\displaystyle{1\over\sqrt{-g}}~{}g_{\mu\lambda}~{}{\delta\mathcal{L}_{GB}\over\delta
g_{\nu\lambda}}={\alpha(n-2)(n-3)\over
2(n-1)}~{}\mathcal{K}^{2}~{}\delta^{\nu}_{\mu}$ (8)

which is clearly non-vanishing at $n=4$.

To arrive at the reduced field equations we consider an spacetime of the form
in $n-$dimensions

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{\nu}dt^{2}+e^{\lambda}dr^{2}+r^{2}d\Omega^{2}_{n-2}.$ (9)

Here, $d\Omega^{2}_{n-2}$ represents the $n-2$ dimensional surface of a unit
sphere. Further, assuming stress tensor for anisotropic fluid as

$\displaystyle\mathcal{T}_{\mu\nu}=(\rho+p_{\theta})u_{\mu}u_{\nu}+p_{\theta}g_{\mu\nu}+(p_{r}-p_{\theta})\chi_{\nu}\chi^{\mu},$
(10)

where all the symbols have their usual meanings.

Now, the field equations in the limit $n\rightarrow 4$ takes the form

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle{e^{-\lambda}\lambda^{\prime}\over
r}\left[1+{2\alpha(1-e^{-\lambda})\over r^{2}}\right]+{1-e^{-\lambda}\over
r^{2}}$ (11) $\displaystyle\hskip
99.58464pt\left[1-{\alpha(1-e^{-\lambda)}\over r^{2}}\right],$ $\displaystyle
8\pi p_{r}$ $\displaystyle=$ $\displaystyle{e^{-\lambda}\nu^{\prime}\over
r}\left[1+{2\alpha(1-e^{-\lambda})\over r^{2}}\right]-{1-e^{-\lambda}\over
r^{2}}$ (12) $\displaystyle\hskip
99.58464pt\left[1-{\alpha(1-e^{-\lambda)}\over r^{2}}\right],$ $\displaystyle
8\pi p_{\theta}$ $\displaystyle=$ $\displaystyle{e^{-\lambda}\over
4}\bigg{[}(2\nu^{\prime\prime}+\nu^{\prime
2})\left\\{1+{4\alpha(1-e^{-\lambda})\over r^{2}}\right\\}+$ (13)
$\displaystyle{2(\nu^{\prime}-\lambda^{\prime})\over
r}~{}\left\\{1-{2\alpha(1-e^{-\lambda})\over
r^{2}}\right\\}-\lambda^{\prime}\nu^{\prime}$ $\displaystyle\hskip
28.45274pt\left\\{1-{8\alpha\over r^{2}}+{12\alpha(1-e^{-\lambda})\over
r^{2}}\right\\}\bigg{]}$ $\displaystyle\hskip
85.35826pt-{2\alpha(1-e^{-\lambda})^{2}\over r^{4}}.$

Since the above three field equations contain 5 unknowns, we need to assume
two variables. The method of solving is considered in the next section.

##  3 Quark matter and the MIT bag model

The Lagrangian in MIT bag model can be written as

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\Big{[}{i\over
2}\Big{\\{}\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-(\partial_{\mu}\overline{\psi})\gamma^{\mu}\psi\Big{\\}}-\mathcal{B}\Big{]}\Theta(x)-{1\over
2}\overline{\psi}\psi\Delta_{s}$

where the symbols have their usual meanings. The Heaviside function and its
derivative is defined as

$\displaystyle\Theta_{\nu}(x)=\Theta(\mathcal{R}-r)~{}~{}~{}\mbox{and}~{}~{}~{}\partial_{\mu}\Theta=n_{\mu}\Delta_{s},$
(15)

Here $\mathcal{R}$ is the radius of the bag, inside which all the three quarks
$u,~{}d,~{}s$ are able to move freely and $n_{\mu}$ is the unit vector normal
to surface. The corresponding energy-momentum tensor of (3) can be determined
as

$\displaystyle T^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[{\partial\mathcal{L}\over\partial(\partial_{\mu}\psi)}~{}\partial^{\nu}\psi+(\partial^{\nu}\overline{\psi})~{}{\partial\mathcal{L}\over\partial(\partial_{\mu}\overline{\psi})}\right]-\mathcal{L}\,g^{\mu\nu}$
(16)

which will eventually become

$\displaystyle T^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\mathcal{L}\,g^{\mu\nu}+{i\over
2}\big{\\{}\overline{\psi}\gamma^{\mu}\partial^{\nu}\psi-(\partial^{\nu}\overline{\psi})\gamma^{\mu}\psi\big{\\}}\Theta.$
(17)

The conservation of energy and momentum suggests

$\displaystyle\partial_{\mu}T^{\mu\nu}=\partial_{\mu}(g^{\mu\nu}\mathcal{B}\,\Theta+{1\over
2}~{}g^{\mu\nu}\overline{\psi}\psi\Delta_{s})=0.$ (18)

In flat space i.e Minkowski space $\partial_{\mu}g^{\mu\nu}=\partial^{\nu}$,
therefore

$\displaystyle\partial^{\nu}(\mathcal{B}\,\Theta)+{1\over
2}~{}\partial^{\nu}(\overline{\psi}\psi\Delta_{s})=0$ or
$\displaystyle-\mathcal{B}~{}n^{\nu}\Delta_{s}+{1\over
2}~{}\partial^{\nu}(\overline{\psi}\psi)\Delta_{s}=0$ or
$\displaystyle\mathcal{B}~{}n^{\nu}\Delta_{s}={1\over
2}~{}\partial^{\nu}(\overline{\psi}\psi)\Delta_{s}.$ (19)

Here we have used $\partial^{\nu}\Theta=-n^{\nu}\Delta_{s}$ and
$\partial^{\nu}\Delta_{s}=0$. The $\Delta_{s}$ function is only non-vanishing
at $r=\mathcal{R}$, hence the bag constant on the surface is

$\mathcal{B}=-{1\over 2}~{}n_{\nu}\,\partial^{\nu}(\overline{\psi}\psi)$ (20)

as $n^{\nu}$, a spacelike vector. Further, the bag is considered spherically
symmetric and one can write $n_{\nu}\partial^{\nu}=d/dr$, therefore

$\mathcal{B}=-{1\over 2}{\partial(\overline{\psi}\psi)\over dr}.$ (21)

This means that the outward pressure due to the confined quarks on the surface
of the bag is counter balancing with the inward QCD vacuum pressure
$\mathcal{B}$.

The pressure, energy density, baryon number density and entropy at finite
temperature for quarks Fermi gas of mass $m_{f}$ and chemical potential
$\mu_{f}$ are

$\displaystyle p$ $\displaystyle=$ $\displaystyle\sum_{f}^{u,d,s}{1\over
3}{g_{f}\over 2\pi^{2}}\int_{0}^{\infty}\kappa{\partial
E_{f}(\kappa)\over\partial\kappa}\Big{[}n(\kappa,\mu_{f})+n(\kappa,-\mu_{f})\Big{]}\kappa^{2}d\kappa$
(22) $\displaystyle\hskip 170.71652pt-\mathcal{B},$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle\sum_{f}^{u,d,s}{g_{f}\over
2\pi^{2}}\int_{0}^{\infty}E_{f}(\kappa)\Big{[}n(\kappa,\mu_{f})+n(\kappa,-\mu_{f})\Big{]}\kappa^{2}d\kappa+\mathcal{B},$
$\displaystyle\rho_{B}$ $\displaystyle=$ $\displaystyle\sum_{f}^{u,d,s}{1\over
3}{g_{f}\over
2\pi^{2}}\int_{0}^{\infty}\Big{[}n(\kappa,\mu_{f})-n(\kappa,-\mu_{f})\Big{]}\kappa^{2}d\kappa,$
(24) $\displaystyle S$ $\displaystyle=$ $\displaystyle\left.{\partial
p\over\partial T}\right|_{V,\mu_{f}},$ (25)

where the kinetic energy of the quarks are

$E_{f}(\kappa)=\sqrt{m_{f}^{2}+\kappa^{2}}$ (26)

and $n(\kappa,\mu_{f})$ the Fermi distribution function

$n(\kappa,\pm\mu_{f})={1\over\exp\big{[}\\{E_{f}(\kappa)\pm\mu_{f}\\}/T\big{]}+1}$
(27)

with $T$ the temperature and $g_{f}=2_{spin}\otimes 3_{color}$ . Under the
massless quarks approximation i.e. $m_{f}=0$, the pressure $p$ and the energy
density $\rho$ are link by a linear equation of state (EoS)

$\displaystyle\hskip 14.22636ptp=\frac{1}{3}\Big{(}\rho-4\mathcal{B}\Big{)},$
(28)

which is the famous MIT-Bag EoS.

##  4 Quark star solution in MIT bag model

Following the assumption of the MIT bag EOS (28), we consider the assumed MIT
bag equation of state (EoS) corresponding to radial presure together with a
well-behaved ansatz for the metric potential $g_{rr}$ to solve the field
equations as,

$\displaystyle\hskip 14.22636ptp_{r}={1\over
3}\big{(}\rho-4\mathcal{B}\big{)},$ (29) $\displaystyle\hskip
14.22636pte^{\lambda}=1+ar^{2}+br^{4},$ (30)

where $\mathcal{B}$ is a bag constant, while $a$ and $b$ are arbitrary
constants with dimension $l^{2}$ and $l^{-4}$, respectively. This form of the
metric function is inspired due to the properties $e^{\lambda(0)}=1$ and
increasing function of $r$.

Then the Eqs.(11), (12), (29) and (30) yields the metric potential $\nu$ as,

$\displaystyle\nu(r)={1\over 9}\bigg{[}3r^{4}(-8\pi a\mathcal{B}+16\pi\alpha
b\mathcal{B}+b)+6r^{2}\big{\\{}16\pi\alpha a\mathcal{B}+$ $\displaystyle\hskip
14.22636pta-3\alpha
b-8\mathcal{B}\left(4\pi\alpha^{2}b+\pi\right)\big{\\}}+3\log\left(ar^{2}+br^{4}+1\right)$
$\displaystyle\hskip 14.22636pt-16\pi
b\mathcal{B}r^{6}-\frac{3\alpha(32\pi\alpha\mathcal{B}+3)}{\sqrt{f_{1}(r)}}\Big{\\{}a^{2}+a\Big{(}\sqrt{f_{1}(r)}-4\alpha
b\Big{)}$ $\displaystyle\hskip
14.22636pt+2b\left(2\alpha^{2}b-\alpha\sqrt{f_{1}(r)}-1\right)\Big{\\}}\log\Big{\\{}a+2b\left(\alpha+r^{2}\right)$
$\displaystyle\hskip
14.22636pt-\sqrt{f_{1}(r)}\Big{\\}}+\frac{3\alpha(32\pi\alpha\mathcal{B}+3)}{\sqrt{f_{1}(r)}}\Big{\\{}a^{2}-a\left(4\alpha
b+\sqrt{f_{1}(r)}\right)$ $\displaystyle\hskip
14.22636pt+2b\left(2\alpha^{2}b+\alpha\sqrt{f_{1}(r)}-1\right)\Big{\\}}\log\Big{\\{}a+2b\left(\alpha+r^{2}\right)$
$\displaystyle\hskip 14.22636pt+\sqrt{f_{1}(r)}\Big{\\}}\bigg{]}+A,$ (31)

where $A$ is the constant of integration and $f_{1}(r)=a^{2}-4\alpha
ab+4b\left(\alpha^{2}b-1\right)$.

Figure 1: Variation metric potentials with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$ and matching of boudary
at the surface.

Now the field equations finally give the expression for $\rho$, $p_{r}$,
$p_{t}$, and $\Delta$ as,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\left(ar^{2}+br^{4}+1\right)^{-3}}{8\pi}\Big{[}\left(a+br^{2}\right)\left(ar^{2}+br^{4}+1\right)$
(32)
$\displaystyle\big{\\{}-\alpha\left(a+br^{2}\right)+ar^{2}+br^{4}+1\big{\\}}+2\left(a+2br^{2}\right)$
$\displaystyle\big{\\{}a\left(2\alpha+r^{2}\right)+b\left(r^{4}+2\alpha
r^{2}\right)+1\big{\\}}\Big{]},$ $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle{1\over 3}\Big{(}\rho-4\mathcal{B}\Big{)},$ (33)
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{4\mathcal{B}}{3}-\frac{1}{24\pi
e^{3\lambda}}\Big{[}e^{\lambda}\left(a+br^{2}\right)\left\\{e^{\lambda}-\alpha\left(a+br^{2}\right)\right\\}+2$
(34) $\displaystyle\left(a+2br^{2}\right)\left(2a\alpha+2\alpha
br^{2}+e^{\lambda}\right)\Big{]}+\frac{1}{8\pi}\bigg{[}-\frac{2\alpha\left(a+br^{2}\right)^{2}}{e^{2\lambda}}$
$\displaystyle+\frac{f_{6}(r)\big{\\{}f_{4}(r)+f_{5}(r)\big{\\}}}{3e^{4\lambda}\big{\\{}a\left(2\alpha+r^{2}\right)+b\left(r^{4}+2\alpha
r^{2}\right)+1\big{\\}}}-\frac{2f_{7}(r)}{3e^{3\lambda}}$
$\displaystyle\frac{a\left(r^{2}-2\alpha\right)+b\left(r^{4}-2\alpha
r^{2}\right)+1}{a\left(2\alpha+r^{2}\right)+b\left(r^{4}+2\alpha
r^{2}\right)+1}+\frac{1}{4e^{\lambda}}\bigg{\\{}\frac{4\alpha\left(a+br^{2}\right)}{e^{\lambda}}$
$\displaystyle+1\bigg{\\}}\bigg{\\{}\frac{4r^{2}\left(f_{8}(r)+f_{9}(r)\right){}^{2}}{9e^{2\lambda}\left(a\left(2\alpha+r^{2}\right)+b\left(r^{4}+2\alpha
r^{2}\right)+1\right)^{2}}+{2\over 9}$
$\displaystyle\bigg{[}f_{10}(r)-\frac{12\left(ar+2br^{3}\right)^{2}}{e^{2\lambda}}+\frac{6\left(a+6br^{2}\right)}{e^{\lambda}}-480\pi
bBr^{4}$ $\displaystyle+\frac{48\alpha b^{2}r^{2}(32\pi\alpha
B+3)f_{2}(r)}{\sqrt{f_{1}(r)}\left\\{a+2b\left(\alpha+r^{2}\right)-\sqrt{f_{1}(r)}\right\\}^{2}}-$
$\displaystyle\frac{12\alpha b(32\pi\alpha
B+3)f_{2}(r)}{\sqrt{f_{1}(r)}\left\\{a+2b\left(\alpha+r^{2}\right)-\sqrt{f_{1}(r)}\right\\}}$
$\displaystyle-\frac{48\alpha b^{2}r^{2}(32\pi\alpha
B+3)f_{3}(r)}{\sqrt{f_{1}(r)}\left\\{a+2b\left(\alpha+r^{2}\right)+\sqrt{f_{1}(r)}\right\\}^{2}}$
$\displaystyle+\frac{12\alpha b(32\pi\alpha
B+3)f_{3}(r)}{\sqrt{f_{1}(r)}\left\\{a+2b\left(\alpha+r^{2}\right)+\sqrt{f_{1}(r)}\right\\}}\bigg{]}\bigg{\\}}\bigg{]},$
$\displaystyle p_{\theta}$ $\displaystyle=$ $\displaystyle\Delta+p_{r},$ (35)

Figure 2: Variation energy density with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$. Figure 3: Variation
pressure with $r$ for $M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$.
Figure 4: Variation pressure anisotropy with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$.

where,

$\displaystyle f_{2}(r)$ $\displaystyle=$ $\displaystyle
a^{2}+a\left(\sqrt{f_{1}(r)}-4\alpha
b\right)+2b\left(2\alpha^{2}b-\alpha\sqrt{f_{1}(r)}-1\right)$ $\displaystyle
f_{3}(r)$ $\displaystyle=$ $\displaystyle a^{2}-a\left(4\alpha
b+\sqrt{f_{1}(r)}\right)+2b\left(2\alpha^{2}b+\alpha\sqrt{f_{1}(r)}-1\right)$
$\displaystyle f_{4}(r)$ $\displaystyle=$ $\displaystyle
2a^{3}r^{2}\left(\alpha+8\pi Br^{4}-r^{2}\right)+a^{2}r^{2}\big{\\{}48\pi
Br^{2}\left(br^{4}+1\right)$ $\displaystyle-6br^{4}+6\alpha
br^{2}-5\big{\\}}+16\pi B\left(br^{4}+1\right)^{3}$ $\displaystyle f_{5}(r)$
$\displaystyle=$ $\displaystyle a\big{\\{}-6b^{2}r^{8}+6\alpha
b^{2}r^{6}+48\pi B\left(br^{5}+r\right)^{2}-11br^{4}-3$ $\displaystyle-2\alpha
br^{2}\big{\\}}-2br^{2}\left\\{b^{2}\left(r^{8}-\alpha
r^{6}\right)+br^{2}\left(\alpha+3r^{2}\right)+2\right\\}$ $\displaystyle
f_{6}(r)$ $\displaystyle=$
$\displaystyle\left(a+2br^{2}\right)\left\\{r^{4}(a+4\alpha
b)+r^{2}(4a\alpha+1)-8\alpha+br^{6}\right\\}$ $\displaystyle f_{7}(r)$
$\displaystyle=$ $\displaystyle a^{3}\left(\alpha
r^{2}-r^{4}\right)-a^{2}\left(-3\alpha+3br^{6}-3\alpha br^{4}+r^{2}\right)-$
$\displaystyle abr^{2}\left(-8\alpha+3br^{6}-3\alpha
br^{4}+r^{2}\right)+br^{2}\big{\\{}b^{2}\left(\alpha r^{6}-r^{8}\right)$
$\displaystyle+5\alpha br^{2}+1\big{\\}}+8\pi Be^{3\lambda}$ $\displaystyle
f_{8}(r)$ $\displaystyle=$ $\displaystyle 2a^{3}r^{2}\left(\alpha+8\pi
Br^{4}-r^{2}\right)+a^{2}r^{2}\big{\\{}48\pi Br^{2}\left(br^{4}+1\right)$
$\displaystyle-6br^{4}+6\alpha br^{2}-5\big{\\}}+16\pi
B\left(br^{4}+1\right)^{3}$ $\displaystyle f_{9}(r)$ $\displaystyle=$
$\displaystyle a\big{\\{}-6b^{2}r^{8}+6\alpha b^{2}r^{6}+48\pi
B\left(br^{5}+r\right)^{2}-11br^{4}-3$ $\displaystyle-2\alpha
br^{2}\big{\\}}-2br^{2}\left\\{b^{2}\left(r^{8}-\alpha
r^{6}\right)+br^{2}\left(\alpha+3r^{2}\right)+2\right\\}$ $\displaystyle
f_{10}(r)$ $\displaystyle=$ $\displaystyle 12\left\\{16\pi\alpha aB+a-3\alpha
b-8B\left(4\pi\alpha^{2}b+\pi\right)\right\\}+36r^{2}$ $\displaystyle(b-8\pi
aB+16\pi\alpha bB)$

Figure 5: Variation equation of state parameter with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$. Figure 6: Variation
sound speed with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$.

The rest of the physical parameters can be deduce from the density and
pressures. The equation of state parameters i.e. $\omega_{r}=p_{r}/\rho$ and
$\omega_{\theta}=p_{\theta}/\rho$ must be less than unity for all physical
matter and unity for Zeldovich fluid ($p_{z}=\rho_{z}$).

Figure 7: Variation adiabatic index with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$.

##  5 Boundary conditions

To show the continuity for the interior spacetime to the external one, one
need boundary matching conditions. The exterior solution is given by Glavan
and Lin glavan2020einstein  in the limit $n\rightarrow 4$ as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-F(r)dt^{2}+{dr^{2}\over
F(r)}+r^{2}d\Omega_{2}^{2},$ (36)

where,

$\displaystyle F(r)$ $\displaystyle=$ $\displaystyle 1+{r^{2}\over
32\pi\alpha}\left[1\pm\left\\{1+{128\pi\alpha M\over
r^{3}}\right\\}^{1/2}\right].$ (37)

The above exterior spacetime has no meaningful solution at short distances if
$\alpha<0$, however, for $\alpha>0$ has two branches of solution. The
asymptotic nature (at $r\rightarrow\infty$) of the two branches (’$-$’ or
’$+$’) for $\alpha>0$ are

$\displaystyle F(r)\rightarrow 1-{2M\over r}~{}~{}\mbox{or}~{}~{}1+{r^{2}\over
16\pi\alpha}+{2M\over r}.$ (38)

The negative branch in (37) asymptotically coincides with Schwarzschild
exterior and hence we will choose

$\displaystyle F(r)$ $\displaystyle=$ $\displaystyle 1+{r^{2}\over
32\pi\alpha}\left[1-\left\\{1+{128\pi\alpha M\over
r^{3}}\right\\}^{1/2}\right].$ (39)

Now matching the interior and exterior spacetime at the boundary $r=R$, we get

$\displaystyle e^{\nu(R)}=e^{-\lambda(R)}=F(R).$ (40)

The this boundary condition, we get

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1-\left(bR^{4}+1\right)F(R)}{R^{2}F(R)},$ (41)
$\displaystyle A$ $\displaystyle=$ $\displaystyle\log[F(R)]-{1\over
9}\bigg{[}3R^{4}(-8\pi a\mathcal{B}+16\pi\alpha
b\mathcal{B}+b)+6R^{2}\big{\\{}16\pi$ (42) $\displaystyle\alpha
a\mathcal{B}+a-3\alpha
b-8\mathcal{B}\left(4\pi\alpha^{2}b+\pi\right)\big{\\}}+3\log(aR^{2}+bR^{4}+1)$
$\displaystyle-16\pi
b\mathcal{B}R^{6}-\frac{3\alpha(32\pi\alpha\mathcal{B}+3)}{\sqrt{f_{1}(R)}}\Big{\\{}a^{2}+a\Big{(}\sqrt{f_{1}(R)}-4\alpha
b\Big{)}$
$\displaystyle+2b\left(2\alpha^{2}b-\alpha\sqrt{f_{1}(R)}-1\right)\Big{\\}}\log\Big{\\{}a+2b\left(\alpha+R^{2}\right)$
$\displaystyle-\sqrt{f_{1}(R)}\Big{\\}}+\frac{3\alpha(32\pi\alpha\mathcal{B}+3)}{\sqrt{f_{1}(R)}}\Big{\\{}a^{2}-a\left(4\alpha
b+\sqrt{f_{1}(R)}\right)$
$\displaystyle+2b\left(2\alpha^{2}b+\alpha\sqrt{f_{1}(R)}-1\right)\Big{\\}}\log\Big{\\{}a+2b\left(\alpha+R^{2}\right)$
$\displaystyle+\sqrt{f_{1}(R)}\Big{\\}}\bigg{]}$

with $f_{1}(R)=a^{2}-4\alpha ab+4b\left(\alpha^{2}b-1\right)$. Further, to
define the surface of the compact structure the pressure must vanishes at the
surface where the two spacetime met i.e. at $r=R$ and therefore $p_{r}(R)=0$,
we get

$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi\left(aR^{2}+bR^{4}+1\right)^{3}}\Big{[}a^{3}(R^{4}-\alpha
R^{2})+a^{2}\big{\\{}3\alpha+$ (43) $\displaystyle 3bR^{6}-3\alpha
bR^{4}+4R^{2}\big{\\}}+a\big{\\{}3b^{2}\left(R^{8}-\alpha R^{6}\right)+10b$
$\displaystyle
R^{2}\left(\alpha+R^{2}\right)+3\big{\\}}+bR^{2}\big{\\{}b^{2}(R^{8}-\alpha
R^{6})+b(6R^{4}+$ $\displaystyle 7\alpha R^{2})+5\big{\\}}\Big{]}~{}.$

The surface of the quark star will be use to determine the bag constant and
compare with the suggested range from the quantum chromodynamical prediction.

##  6 Physical properties of the solution

The obtained solution must be tested with several physical constraints to show
its physically acceptability. The variation of the metric function is shown in
Fig. 1. The meeting of the two metric functions at the surface verify the
satisfaction of boundary conditions. The non-increasing nature of the energy
density is shown in Fig. 2 while the decreasing nature of the pressures is
shown in Fig. 3 where the radial pressure vanishes at the surface. Further,
the anisotropy must be vanishing at the center for well-behaved solution. Fig.
4 shows the nature of the anisotropy, which is positive throughout the star.
This implies that the anisotropic force is directed outward which helps to
prevents the gravitational collapse. Now one must also verify if the equation
of state (EoS) parameter is below that of Zeldovich’s fluid. This is because
Zeldovich’s fluid is the most stiff EoS known till date with
$\omega=p/\rho=1$. Fig. 5 shows that the EoS parameters along the radial as
well as tangential directions are below 1 i.e. it represents a physical fluid
system. Causality condition is one of the most important test to check a new
theory/solution whether physical or not. This requires that the speed of sound
in system must be subluminal i.e. less than that of light. For the solution
the speed of sound can be found as

$\displaystyle\hskip 14.22636ptv^{2}_{r}={dp_{r}\over
d\rho}~{}~{}~{}~{}\&~{}~{}~{}v_{\theta}^{2}={dp_{\theta}\over d\rho}.$ (44)

The variation of the speed of sound yielded from the solution is subluminal
(see Fig 6). Further, the ratio of the two specific indices i.e.
$\gamma=C_{p}/C_{v}$ is an important parameter that determined a stellar
system stable or not. For a Newtonian stellar fluid $\gamma>4/3$ or otherwise
will initiate gravitational collapse. However, for general relativistic fluids
which are generally anisotropic, deviated from the Newtonian case. For
relativistic fluid the adiabatic index is given by

$\displaystyle\hskip 14.22636pt\gamma={\rho+p_{r}\over p_{r}}~{}{dp_{r}\over
d\rho}.$ (45)

Here, the collapse limit depends on the nature of the anisotropy. For
$p_{r}>p_{\theta}$ can even have $\gamma<4/3$ and still stable or for
$p_{\theta}>p_{r}$ can be unstable even if $\gamma>4/3$. For this solution it
is clear that the central values of the adiabatic index (minimum for
$\alpha=0$ at 2.8) is far above 4/3 (see Fig. 7) and hence the solution is
free from gravitational collapses.

Figure 8: Variation mass with central density for $b=0.000015/km^{4}$. Figure
9: Variation energy conditions with $r$ for
$M=1.97M_{\odot},~{}R=9.69~{}km,~{}b=0.000015/km^{4}$.

All the above physical analyses clearly indicate that the solution is
physically acceptable. However, one questions remains, is the system able to
sustain stability under radial perturbations? This answer lies with static
stability criterion. This criterion signifies that whenever the mass of a
stellar fluid is an increasing function of its central density i.e.
$dM/d\rho_{c}>0$ then system sustain stability under radial perturbations zel
; har . The mass is given by

$\displaystyle M(\rho_{c})$ $\displaystyle=$
$\displaystyle\frac{R^{3}}{128\pi\alpha}\bigg{[}\bigg{\\{}1+\frac{32\pi\alpha}{R^{2}}$
(46) $\displaystyle\hskip
42.67912pt\left(1-\frac{1}{aR^{2}+bR^{4}+1}\right)\bigg{\\}}^{2}-1\bigg{]}$
$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{\sqrt{768\pi^{2}\alpha\rho_{c}+9}-3}{6\alpha}.$ (47)

The variation of mass with respect to its central density in Fig. 8 shows that
the solution fulfilled the static stability criterion. Finally, we must also
needed to check whether the stellar fluid is physical or not using the energy
conditions. For any physical fluids, the weak ($\rho\geq 0,~{}\rho+p_{i}\geq
0$), null ($\rho+p_{i}\geq 0$), dominant ($\rho\geq|p_{i}|$) and strong
($\rho+\sum p_{i}$) energy conditions must satisfy. Figure 9 shows the
satisfaction of these energy conditions.

Figure 10: Range of bag constant corresponding to different $M$ and $\alpha$ for $R=9.69km,~{}b=0.000015/km^{4}$. The black lines are equi-$\mathcal{B}$ in $MeV/fm^{3}$. Figure 11: Range of bag constant corresponding to different $M$ and $\alpha$ for $R=12km,~{}b=0.000015/km^{4}$.The black lines are equi-$\mathcal{B}$ in $MeV/fm^{3}$. Figure 12: $\mathcal{B}-\alpha$ plane for equi-$M/M_{\odot}$ for $R=9.69km,~{}b=0.000015/km^{4}$. Table 1:  Values of all the parameters corresponding to different values of $\alpha$ and for $M=1.97M_{\odot},~{}R=9.69km$. $\alpha$ |  $b\times$ 10-5 |  $\rho_{s}\times$ 1014 |  $p_{c}\times$ 1035 | $\mathcal{B}$ | $z_{s}$
---|---|---|---|---|---
$(km^{2})$ | $(km^{-4})$ | ($g/cm^{3}$) | ($dyne/cm^{2}$) | $(MeV/fm^{3})$ |
0 | $1.5$ | 5.576 | 1.075 | 89.85 | 0.290
0.06 | $1.5$ | 5.515 | 0.960 | 89.51 | 0.280
0.12 | $1.5$ | 5.455 | 0.854 | 89.15 | 0.270
0.18 | $1.5$ | 5.406 | 0.761 | 88.79 | 0.263
0.24 | $1.5$ | 5.346 | 0.676 | 88.44 | 0.255

On the other hand, the predicted range of $\mathcal{B}$ for $R=9.69\,km$ and
$12\,km$ with the same mass range is shown in Figs. 10 and 11. These figures
show the ranges of $\mathcal{B}$ by varying the mass $M$ and Gauss-Bonnet
coupling constant $\alpha$. As can be seen that for same mass and radius the
bag constant decreases with increase in coupling constant $\alpha$. Also, a
high mass quark star configuration is possible for low $\alpha$ and high
$\mathcal{B}$ or high $\alpha$ and low $\mathcal{B}$. Further, the ranges of
bag constant estimated for $0\leq\alpha\leq 3$ and $1M_{\odot}\leq M\leq
3M_{\odot}$ for 9.69 km radius is $45MeV/fm^{3}\leq\mathcal{B}\leq
80MeV/fm^{3}$ and for 12 km radius is $40MeV/fm^{3}\leq\mathcal{B}\leq
56MeV/fm^{3}$. The predicted range of bag constant lies between $57MeV/fm^{3}$
and $92MeV/fm^{3}$ bla . This clear means that, for example, a quark star of
mass $2M_{\odot}$ with radius 12 km (See Fig. 11) is highly unlikely because
in such case the bag constant should be in range
$46MeV/fm^{3}<\mathcal{B}<50MeV/fm^{3}$ which well below the predicted range.
The $\mathcal{B}-\alpha$ plan with equi-mass lines are also plotted in Fig. 12
for the ranges $0\leq\alpha\leq 2$ and $57MeV/fm^{3}\leq\mathcal{B}\leq
78MeV/fm^{3}$. This plot clearly shows that the higher masses of a quark star
configuration is possible for low bag constant and higher Gauss-Bonnet
coupling constant. This also implies that reducing the bag constant make the
EoS stiff with the increase of $\alpha$-coupling. Table 1 shows how the
physical parameters such as surface density, central pressure, surface
redshift and bag constant varies with the Gauss-Bonnet coupling strength for a
particle stellar system of mass $1.97M_{\odot}$ and radius 9.69 $km$.

##  7 Results and discussions

We have successfully discover the first exact solution describing quark stars
in 4-$D$ EGB gravity. This solution match with the exterior solution smoothly
at the surface of the star $r=R$ (Fig. 1), which has non-increasing energy
density & pressure (Figs. 2 & 3) as well as vanishing anisotropy at $r=0$
(Fig. 4). For this solution, the EoS parameters are less than unity (Fig. 5)
and satisfy the causality condition (Fig. 6). Moreover, the solution has been
also tested with stringent physical constraints like Bondi criterion, static
stability criterion and energy conditions. As per the central values of
adiabatic index (see Fig. 7) and the static stability criterion (see Fig. 8),
one can clearly see the $\gamma_{c}$ is more and more stable density range for
higher coupling constant $\alpha$. This only implies that the quark star
configuration is more stable with larger coupling strength i.e. when the
Gauss-Bonnet term is contribution more into the field equations. Moreover, the
solution also fulfilled all the energy conditions which was required for any
physical system (Fig. 9). The interior solution was also matched with the
Schwarzschild-like exterior spacetime. One of these matching conditions was
also used to predicts the range of bag constant $\mathcal{B}$ without any
concepts of microscopic QCD calculations. The $\alpha-M$ contour graphs in
Figs. 10 and 11 shows that which configurations of quark stars are physically
motivated keeping in mind the predicted range of $\mathcal{B}$ as per QCD
calculations. The $\mathcal{B}-\alpha$ plot with equi-mass contours clearly
indicate that quark star mass gets heavier with decreasing bag constant and
increasing $\alpha$ coupling. The present work has three significant results
(i) it might have astrophysical interest in future, (ii) a way of predicting
the range of bag constant in high energy density regime and (iii) can
determine the mass of quark star configuration for wide ranges of $\alpha$ and
$\mathcal{B}$.

To see if the 4D EGB gravity reduces to GR under the condition
$\alpha\rightarrow 0$, one has to see its vacuum solution. The vacuum solution
in (37) at $\alpha\rightarrow 0$ reduces to

$\displaystyle F(r)\approx 1+{r^{2}\over
32\pi\alpha}\Big{[}1\pm\Big{\\{}1+{1\over 2}\,{128\pi\alpha M\over
r^{3}}+\text{O}(\alpha^{2})\Big{\\}}\Big{]}.$ (48)

Hence, the negative branch of the above function reduces to

$\displaystyle F(r)\rightarrow 1-{2M\over r},$ (49)

which is again the Schwarzschild vacuum solution. However, the positive branch
reduces to

$\displaystyle F(r)\approx 1+{r^{2}\over 32\pi\alpha}\Big{[}2+{1\over
2}\,{128\pi\alpha M\over r^{3}}+\text{O}(\alpha^{2})\Big{]},$ (50)

which is undefined at $\alpha\rightarrow 0$. Therefore, the negative branch of
the vacuum solution coincides with the Schwarzschild’s vacuum solution under
$\alpha\rightarrow 0$ and $r\rightarrow\infty$ conditions while the positive
branch doesn’t coincide under these two conditions. Therefore, only the
negative branch makes it more physical and the 4D EGB gravity can reduce to
GR. However, both the negative and positive branches are solutions of 4D EGB
gravity hence it cannot completely reduces to GR at $\alpha\rightarrow 0$.
Further, since the positive branch blows-up both at $r\rightarrow\infty$ and
$\alpha\rightarrow 0$, this solution is physically meaningless at least in
4D-gravity.

###### Acknowledgements.

Authors are grateful to Prof. Sushant Ghosh, Centre for Theoretical Physics,
Jamia Millia Islamia, New Delhi for useful suggestions on improving the
contents of the paper. The author S. K. Maurya acknowledge that this work is
carried out under TRC project-BFP/RGP/CBS/19/099 of the Sultanate of Oman. The
author also acknowledges to University of Nizwa administration for their
continuing support and encouragements. FR would like to thank the authorities
of the Inter-University Centre for Astronomy and Astrophysics, Pune, India for
providing research facilities. This work is a part of the project submitted by
FR in DST-SERB, Govt. of India.

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