Abstract.

In this paper, we introduce and investigate the notions of $cs$-regular, $sn$-regular, $so$-regular, and regular families at arbitrary subsets of topological spaces. We study their relationships with point-2-star, $cs$-star, $sn$-star, $so$-star, and $o$-star networks, as well as with $cs$-metrizable spaces and images of metric spaces under $k$-$cs$ and sequence-covering mappings at such subsets. Moreover, several related concepts are introduced, which enable us to establish new results and to recover, as special cases, some results previously obtained by S. Lin, W.P. Zheng, and Z.Y. Cai (2020).

1. Introduction

The point-regular (resp., $cs$-regular, regular) covers of topological spaces play an important role not only in metrization theory but also in the mutual classification of mappings and spaces (see [1, 2, 4, 5, 9, 15, 16, 22, 25]). In particular, in [2, 5, 9, 22], the authors investigated spaces with a point-regular base (resp., weak base, $sn$-network, $cs$-network, $c{s}^{\ast }$-network), and examined their relationships with various classes of spaces, including metrizable spaces, spaces admitting a point-star network consisting of point-finite open covers (resp., $sn$-covers, $cs$-covers, $c{s}^{\ast }$-covers), as well as images of metric spaces under certain classes of mappings.

On the other hand, since the set of non-isolated points is a special subset of a topological space, Lin et al. gave some characterizations about point-regular covers at arbitrary subsets of topological spaces (see [20, 21]). More precisely, for a subset $A$ of a space $X$, they proved that $X$ has a point-regular external base (resp., $so$-network, $sn$-network, $cs$-network, $c{s}^{\ast }$-network) at $A$ for $X$ if and only if $X$ has a sequence of point-finite open covers (resp., $so$-covers, $sn$-covers, $cs$-covers, $c{s}^{\ast }$-covers) at $A$ which is a point-star network at $A$ for $X$. Consequently, their results extended some of those obtained in [2, 5, 9, 22].

More recently, in [23], S. Lin, W.P. Zheng, and Z.Y. Cai introduced and systematically studied the notions of $sn$-finite, $so$-finite, $sn$-regular, and $so$-regular families. These notions generalize the classical concepts of point-finite, $cs$-finite, $cs$-regular, regular, and locally finite families, as well as the notions of point-2-star, $cs$-star, and $o$-star networks, which themselves extend the concept of point-star networks. Moreover, the authors introduced the concept of $k$-$cs$-mappings and established the following characterizations.

Theorem 1.1 ([23], Theorems 4.2, 4.6).

The following are equivalent for a space $X.$

1. $X$ is $cs$-metrizable.

2. $X$ has an $so$-regular $so$-network.

3. $X$ has a $cs$-regular $c{s}^{\ast }$-network.

4. $X$ has a sequence of $so$-finite $so$-covers which is a $cs$-star network.

5. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers which is a $cs$-star network.

6. $X$ has a sequence of $so$-finite $so$-covers which is a point-2-star network.

7. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers which is a point-2-star network.

8. $X$ is a $k$-$cs$-(and sequence-covering) image of a metric space.

9. $X$ has a $\sigma$-$cs$-finite $so$-network and $sX$ is a regular space.

Theorem 1.2 ([23], Theorem 5.2).

The following are equivalent for a space $X$.

1. $X$ has a regular $so$-network.

2. $X$ has a regular $cs$-network.

3. $X$ has a sequence of regular and $so$-finite $so$-covers which is an $o$-star network.

4. $X$ has a sequence of regular $cs$-covers which is an $o$-star network.

Motivated by [20, 21, 24, 27, 28], in this paper we introduce and systematically study several new notions, including $cs$-finite, $sn$-finite, $so$-finite, $cs$-regular, $sn$-regular, $so$-regular, and regular families, as well as $sn$-star, $so$-star, $o$-star, and point-2-star networks, together with $k$-$cs$-mappings at arbitrary subsets of topological spaces. Our aim is to extend and unify the above-mentioned results in this more general setting. We establish the following main results; see Theorems 3.12 and 3.19.

Theorem 1.3.

Let $A$ be a sequentially open subset of a space $X.$ Then, the following are equivalent.

1. $A$ is $cs$-metrizable.

2. $X$ has an $so$-regular $so$-network at $A$ for $X$.

3. $X$ has a $cs$-regular $c{s}^{\ast }$-network at $A$ for $X$.

4. $X$ has a sequence of $so$-finite $so$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

5. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

6. $X$ has a sequence of $c{s}^{\prime }$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

7. $X$ has a sequence of $so$-finite $so$-covers at $A$ which is a point-2-star network at $A$ for $X$.

8. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers at $A$ which is a point-2-star network at $A$ for $X$.

9. $X$ is an image of a metric space under a $k$-$cs$-(and sequence-covering) mapping at $A.$

10. $X$ has a $\sigma$-$cs$-finite $so$-network at $A$ for $X$ and $sA$ is a regular space.

Theorem 1.4.

Let $A$ be a sequentially open subset of a space $X.$ Then, the following are equivalent.

1. $X$ has a regular $so$-network at $A$ for $X$.

2. $X$ has a regular $cs$-network at $A$ for $X$.

3. $X$ has a sequence of regular and $so$-finite $so$-covers at $A$ which is an $o$-star network at $A$ for $X$.

4. $X$ has a sequence of regular $cs$-covers at $A$ which is an $o$-star network at $A$ for $X$.

As a consequence of the above results, when $A=X$, we recover several known results in [23], including Theorems 4.2, 4.6, and 5.2.

2. Preliminaries

In this section, we give several concepts and some relationships among them. Throughout this paper, all spaces are assumed to be $T_2$, all mappings are continuous and onto, and $\mathbb{N}$ denotes the set of all positive integers.

Let $P$ be a subset of a topological space $X$, and let $x\in X$. Suppose that a sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in $X$ converges to $x$. For $m\in \mathbb{N}$, we denote

$$T{[x_n]}_m=\{x\}\cup \{x_n:n\ge m\}.$$

In particular, $T{[x_n]}_1$ is denoted by $T[x_n]$, which is called a convergent set in $X$. The sequence ${\{x_n\}}_{n\in \mathbb{N}}$ is called eventually in $P$, if there exists $m\in \mathbb{N}$ such that $T{[x_n]}_m\subset P$. The sequence ${\{x_n\}}_{n\in \mathbb{N}}$ is called frequently in $P$, if some subsequence of ${\{x_n\}}_{n\in \mathbb{N}}$ is eventually in $P$. Moreover, a set $P$ is called a sequential neighborhood of $x\in X$ [10], if each sequence ${\{x_n\}}_{n\in \mathbb{N}}$ converging to $x$ is eventually in $P$. A set $P$ is called a sequentially open set of $X$ [11], if $P$ is a sequential neighborhood of each point in $P$. The space $X$ is called a sequential space [9], if each sequentially open subset of $X$ is open. Furthermore, if $𝒫$ is a family of subsets of $X$, $A\subset X$ and $x\in X$, then we denote

$\text{St}(A,𝒫)$ $={\displaystyle \bigcup \{P\in 𝒫:P\cap A\ne \mathrm{\varnothing }\}};$

$\text{St}(x,𝒫)$ $=\text{St}(\{x\},𝒫);$

${\text{St}}^2(x,𝒫)$ $=\text{St}(\text{St}(x,𝒫),𝒫);$

${𝒫|}_A$ $=\{P\cap A:P\in 𝒫\};$

${(𝒫)}_A$ $=\{P\in 𝒫:P\cap A\ne \mathrm{\varnothing }\};$

${(𝒫)}_x$ $={(𝒫)}_{\{x\}}.$

Definition 2.1.

Let $𝒫$ be a family of subsets of a space $X$ and $A\subset X$.

1. $𝒫$ is called a network at a point $x\in X$ [3], if $x\in \bigcap 𝒫$, and for each neighborhood $U$ of $x$ in $X$, there is a $P\in 𝒫$ such that $P\subset U$.

2. $𝒫$ is called a $cs$-network (resp., $c{s}^{\ast }$-network) at $A$ for $X$ [20], if for each $x\in A$, any sequence ${\{x_n\}}_{n\in \mathbb{N}}$ converging to $x\in U$ with $U$ open in $X$, then ${\{x_n\}}_{n\in \mathbb{N}}$ is eventually (resp., frequently) in $P\subset U$ for some $P\in 𝒫$. When $A=X$, $𝒫$ is called a $cs$-network [14] (resp., $c{s}^{\ast }$-network [12]) for $X$.

3. $𝒫$ is called an $sn$-network at a point $x\in X$ [18], if the following are satisfied: (i) $𝒫$ is a network at $x$ in $X$; (ii) if $U,V\in 𝒫$, then $W\subset U\cap V$ for some $W\in 𝒫$; (iii) each element of $𝒫$ is a sequential neighborhood of $x$ in $X$.

4. $𝒫={\bigcup }_{x\in A}{𝒫}_x$ is called an external base [3] (resp., $sn$-network [18], $so$-network [18]) at $A$ for $X$, if ${𝒫}_x$ is a local base (resp., an $sn$-network, an $sn$-network consisting of sequentially open sets) at $x$ in $X$ for each $x\in A$. When $A=X$, $𝒫={\bigcup }_{x\in X}{𝒫}_x$ is called a base [3] (resp., an $sn$-network [18], an $so$-network [18]) for $X$.

Remark 2.2.

External base (at $A$) $\Rightarrow$ $so$-network (at $A$) $\Rightarrow$ $sn$-network (at $A$) $\Rightarrow$ $cs$-network (at $A$) $\Rightarrow$ $c{s}^{\ast }$-network (at $A$).

Definition 2.3.

Let $𝒫$ be a family of subsets of a space $X$ and $A\subset X$.

1. $𝒫$ is called a $c{s}^{\prime }$-cover at $A$ for $X$ [27], if for every sequence ${\{x_n\}}_{n\in \mathbb{N}}$ converging to $x\in A$ in $X$, there are a $P\in 𝒫$ and an $n\in \mathbb{N}$ such that $\{x,x_n\}\subset P$. When $A=X$, $𝒫$ is called a $c{s}^{\prime }$-cover of $X$ [19].

2. $𝒫$ is called a $cs$-cover [29] (resp., $c{s}^{\ast }$-cover [21]) at $A$ for $X$, if every sequence ${\{x_n\}}_{n\in \mathbb{N}}$ converging to $x\in A$ in $X$ is eventually (resp., frequently) in some $P\in 𝒫$. When $A=X$, $𝒫$ is called a $cs$-cover [29] (resp., $c{s}^{\ast }$-cover [17]) of $X$.

3. $𝒫$ is called an $sn$-cover at $A$ for $X$ [22], if each element of $𝒫$ is a sequential neighborhood of some point at $A$ in $X$, and for each $x\in A$, there is a sequential neighborhood $P\in 𝒫$ of $x$ in $X$. When $A=X$, $𝒫$ is called an $sn$-cover of $X$ [22].

4. $𝒫$ is called an open cover (resp., $so$-cover [22]) at $A$ for $X$, if each element of $𝒫$ is an open (resp., a sequentially open) set in $X$ and $A\subset \bigcup 𝒫$. When $A=X$, $𝒫$ is called an open cover (resp., $so$-cover [22]) of $X$.

Remark 2.4.

Open cover (at $A$) $\Rightarrow$ $so$-cover (at $A$) $\Rightarrow$ $sn$-cover (at $A$) $\Rightarrow$ $cs$-cover (at $A$) $\Rightarrow$ $c{s}^{\ast }$-cover (at $A$) $\Rightarrow$ $c{s}^{\prime }$-cover (at $A$).

Definition 2.5.

Let $𝒫$ be a family of subsets of a space $X$ and $A\subset X$.

1. $𝒫$ is called point-finite (resp., point-countable) at $A$ [20], if the family ${(𝒫)}_x$ is finite (resp., countable) for each $x\in A$. When $A=X$, $𝒫$ is called point-finite (resp., point-countable) [9].

2. $𝒫$ is called locally finite (resp., locally countable) at $A$ [28], if for each $x\in A$, there exists an open neighborhood $V$ of $x$ in $X$ such that the family ${(𝒫)}_V$ is finite (resp., countable). When, $A=X$, $𝒫$ is called locally finite (resp., locally countable) [9].

3. $𝒫$ is called $cs$-finite at $A$, if each convergent sequence in $A$ intersects at most finitely many elements of $𝒫$. When $A=X$, $𝒫$ is called $cs$-finite [7].

4. $𝒫$ is called $sn$-finite (resp., $so$-finite) at $A$, if for each $x\in A$, there exists a sequential (resp., sequentially open) neighborhood $O_x$ of $x$ in $X$ such that the family ${(𝒫)}_{O_x}$ is finite. When $A=X$, $𝒫$ is called $sn$-finite (resp., $so$-finite) [23].

5. $𝒫$ is called point-regular at $A$ [20], if for each $x\in A$ and $x\in U$ with $U$ open in $X$, $\{P\in {(𝒫)}_x:P\not\subset U\}$ is finite. When $A=X$, $𝒫$ is called point-regular [1].

6. $𝒫$ is called $cs$-regular at $A$, if for each $x\in A$, $x\in U$ with $U$ open in $X$ and a sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in $A$ converges to $x$, there exists $m\in \mathbb{N}$ such that the family $\{P\in {(𝒫)}_{T{[x_n]}_m}:P\not\subset U\}$ is finite. When $A=X$, $𝒫$ is called $cs$-regular [16].

7. $𝒫$ is called regular (resp., $so$-regular, $sn$-regular) at $A$, if for each $x\in A$ and $x\in U$ with $U$ open in $X$, there exists an open (resp., a sequentially open, a sequential) neighborhood $V$ of $x$ in $X$ such that the family $\{P\in {(𝒫)}_V:P\not\subset U\}$ is finite. When $A=X$, $𝒫$ is called regular [4] (resp., $so$-regular [23], $sn$-regular [23]).

[ >=Stealth, every node/.style=font=, align=center ]

(a1) at (0,2) Locally finite family
(at $A$); (a2) at (4,2) $so$-finite
(at $A$); (a3) at (8,2) $sn$-finite
(at $A$); (a4) at (12,2) $cs$-finite
(at $A$); (a5) at (16,2) point-finite
(at $A$);

(b1) at (0,0) regular family
(at $A$); (b2) at (4,0) $so$-regular
(at $A$); (b3) at (8,0) $sn$-regular
(at $A$); (b4) at (12,0) $cs$-regular
(at $A$); (b5) at (16,0) point-regular
(at $A$);

[->] (a1) – (a2); [->] (a2) – (a3); [->] (a3) – (a4); [->] (a4) – (a5);

[->] (b1) – (b2); [->] (b2) – (b3); [->] (b3) – (b4); [->] (b4) – (b5);

[->] (a1) – (b1); [->] (a2) – (b2); [->] (a3) – (b3); [->] (a4) – (b4); [->] (a5) – (b5);

Figure 1. The relationships among certain point-regular families.

Definition 2.6.

Let ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ be a sequence of families of subsets of a space $X$ and $A\subset X$.

1. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a point-star network at $A$ for $X$ [20], if ${\{\text{St}(x,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x$ in $X$ for each $x\in A$. When $A=X$, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a point-star network for $X$ [22].

2. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a point-2-star network at $A$ for $X$, if ${\{{\text{St}}^2(x,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x$ in $X$ for each $x\in A$. When $A=X$, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a point-2-star network for $X$ [23].

3. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a compact-star network at $A$ for $X$ [27], if for any compact subset $K$ in the subspace $A$ of $X$ and $K\subset U$ with $U$ open in $X$, there exists $n\in \mathbb{N}$ such that $\text{St}(K,{𝒫}_n)\subset U$. When $A=X$, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a compact-star network for $X$ [13].

4. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called an $o$-star network (resp., $so$-star network, $sn$-star network) at $A$ for $X$, if for each $x\in A$ and $x\in U$ with $U$ open in $X$, there exist an open (resp., a sequentially open, a sequential) neighborhood $V$ of $x$ in $X$ and $n\in \mathbb{N}$ such that $\text{St}(V,{𝒫}_n)\subset U$. When $A=X$, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called an $o$-star network [23] (resp., $so$-star network, $sn$-star network) for $X$.

5. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a $cs$-star network at $A$ for $X$ [27], if for any convergent set $T\subset A$ and $T\subset U$ with $U$ open in $X$, there exists $n\in \mathbb{N}$ such that $\text{St}(T,{𝒫}_n)\subset U$. When $A=X$, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is called a $cs$-star network for $X$ [19].

Remark 2.7.

1. The point-star network in this paper is equivalent to the $\sigma$-strong network in [15, p. 237].

2. Compact-star network (at $A$) $\Rightarrow$ $cs$-star network (at $A$) $\Rightarrow$ point-star network (at $A$).

3. $o$-star network (at $A$) $\Rightarrow$ $so$-star network (at $A$) $\Rightarrow$ $sn$-star network (at $A$) $\Rightarrow$ $cs$-star network (at $A$) $\Rightarrow$ point-star network (at $A$).

4. Point-2-star network (at $A$) $\Rightarrow$ point-star network (at $A$).

Definition 2.8 ([6]).

Let $X$ be a space.

1. The sequential coreflection $sX$ of $X$ is the set $X$ endowed with the topology consisting of all sequentially open sets of $X$.

2. $X$ is called $cs$-metrizable, if $sX$ is metrizable.

Remark 2.9 ([6]).

$X$ and $sX$ have the same convergent sequences, the same sequential neighborhoods at a point in the spaces and the same sequentially open sets. So, $sX$ is a sequential space, and $X$ is a sequential space if and only if $sX=X$.

Definition 2.10.

Let $f:X\to Y$ be a mapping and $A\subset Y.$

1. $f$ is called a sequentially quotient mapping at $A$ [24], if whenever ${\{y_n\}}_{n\in \mathbb{N}}$ is a sequence converging to a point $y\in A$ in $Y,$ there exist a convergent sequence ${\{x_i\}}_{i\in \mathbb{N}}$ in $X$ and a subsequence ${\{y_{n_i}\}}_{i\in \mathbb{N}}$ of ${\{y_n\}}_{n\in \mathbb{N}}$ with each $x_i\in f^{-1}(y_{n_i})$. When $A=Y,$ $f$ is called a sequentially quotient mapping [8].

2. $f$ is called a sequence-covering mapping at $A$ [24], if whenever ${\{y_n\}}_{n\in \mathbb{N}}$ is a sequence converging to a point $y\in A$ in $Y,$ there exists a convergent sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in $X$ with each $x_n\in f^{-1}(y_n).$ When $A=Y,$ $f$ is called a sequence-covering mapping [26].

3. $f$ is called a $k$-$cs$-mapping at $A$, if whenever $T$ is a convergent set in $A$, $f^{-1}(T)$ is a compact subset of $X.$ When $A=Y,$ $f$ is called a $k$-$cs$-mapping [23].

Definition 2.11.

Let $(X,d)$ be a metric space, $f:X\to Y$ a mapping, and $A\subset Y$. Then, $f$ is called a $cs$-regular mapping at $A$ [27], if for each convergent set $T\subset A$ and each open neighborhood $U$ of $T$ in $Y,$ we have that

$$d(f^{-1}(T),X\setminus f^{-1}(U))>0.$$

When $A=Y,$ $f$ is called a $cs$-regular mapping [19].

Remark 2.12.

Sequence-covering mapping (at $A$) $\Rightarrow$ sequentially quotient mapping (at $A$).

3. Main results

Let $𝒫$ be a family of subsets of a space $X,$ we denote

$I(X)$ $=\{x\in X:x\text{ is an isolated point of }X\};$

$S(X)$ $=\{x\in X:\{x\}\text{ is a sequentially open set in }X\};$

${𝒫}^m$ $=\{Q\in 𝒫:\text{ If }P\in 𝒫\text{ and }Q\subset P,\text{ then }Q=P\}.$

Lemma 3.1 ([19], Lemma 2.1).

Let $X$ be a first-countable space and $x\in X.$ If $P$ is a sequential neighborhood of $x$, then $x\in \text{Int}P$.

Proposition 3.2.

Let $A$ be a sequentially open subset of a space $X.$ Let ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ be a sequence of $c{s}^{\prime }$-covers at $A$ for $X,$ and suppose that ${𝒫}_{n+1}$ refines ${𝒫}_n$ for every $n\in \mathbb{N}.$ Then, the following statements are equivalent.

1. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is an $so$-star network at $A$ for $X.$

2. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X.$

3. ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a point-2-star network at $A$ for $X.$

Proof 3.3.

$(1)\Rightarrow (2).$ This is obvious.

$(2)\Rightarrow (3).$ Assume that ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is not a point-2-star network at $A$ for $X.$ Then, there exist $x\in A$ and an open neighborhood $U$ of $x$ in $X$ such that ${\text{St}}^2(x,{𝒫}_n)\not\subset U$ for every $n\in \mathbb{N}.$ This implies that for each $n\in \mathbb{N},$ there exist $x_n,y_n\in X$ and $P_n\in {𝒫}_n$ such that $x_n\in P_n\setminus U$ and $y_n\in P_n\cap \text{St}(x,{𝒫}_n).$ Since ${\{\text{St}(x,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x$ in $X,$ the sequence ${\{y_n\}}_{n\in \mathbb{N}}$ converges to $x$ in $X.$ Because $A$ is a sequentially open subset of $X,$ there exists $m\in \mathbb{N}$ such that the sequence ${\{y_n\}}_{n\ge m}$ converges to $x$ in $A$ and $T{[y_n]}_m\subset U.$ Moreover, since ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X$, there exists $k\in \mathbb{N}$ such that $\text{St}(T{[y_n]}_m,{𝒫}_n)\subset U$ for all $n\ge k,$ which implies that

$$x_n\in P_n\subset \text{St}(T{[y_n]}_m,{𝒫}_n)\subset U\text{for all }n\ge \max \{m,k\}.$$

This contradicts the fact that $x_n\notin U$ for every $n\in \mathbb{N}.$ Therefore, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a point-2-star network at $A$ for $X$.

$(3)\Rightarrow (1).$ First, we show that ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X$. Indeed, assume that ${\{x_k\}}_{k\in \mathbb{N}}$ is a sequence converging to $x$ in $A$ and $T[x_k]\subset U,$ where $U$ is an open set in $X.$ Since ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a point-2-star network at $A$ for $X$, there exists $m\in \mathbb{N}$ such that ${\text{St}}^2(x,{𝒫}_m)\subset U.$ By [27, Lemma 3.1(4)], $\text{St}(x,{𝒫}_m)$ is a sequential neighborhood of $x$ in $X,$ and hence there exists $k_m\in \mathbb{N}$ such that $T{[x_k]}_{k_m}\subset \text{St}(x,{𝒫}_m).$ For each since ${\{\text{St}(x_k,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x_k$ in $X,$ there exists $n_k\in \mathbb{N}$ such that $\text{St}(x_k,{𝒫}_n)\subset U$ for all $n\ge n_k.$ Now, let

Then, we have that

$\text{St}(T[x_k],{𝒫}_{n_0})$

Therefore, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of $c{s}^{\prime }$-covers at $A$ which is a $cs$-star network at $A$ for $X$. By [27, Theorem 3.11], $sA$ is metrizable. Moreover, since $\text{St}(x,{𝒫}_m)\cap A$ is a sequential neighborhood of $x$ in $sA,$ by Lemma 3.1, we have that

$$x\in {\text{Int}}_{sA}(\text{St}(x,{𝒫}_m)\cap A).$$

On the other hand, since $A$ is a sequentially open subset of $X,$ we have that ${\text{Int}}_{sA}(\text{St}(x,{𝒫}_m)\cap A)$ is sequentially open in $X.$ Hence,

$$x\in \text{St}({\text{Int}}_{sA}(\text{St}(x,{𝒫}_m)\cap A),{𝒫}_m)\subset {\text{St}}^2(x,{𝒫}_m)\subset U.$$

Thus, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is an $so$-star network at $A$ for $X$.

Lemma 3.4.

Let $f:X\to Y$ be a mapping, $X$ be a metric space and $A\subset Y.$ If $f$ is a $k$-$cs$-mapping at $A$, then $f$ is a sequentially quotient and $cs$-regular mapping at $A.$

Proof 3.5.

Assume that ${\{y_n\}}_{n\in \mathbb{N}}$ is a sequence converging to $y$ in $A.$ Let $T=\{y\}\cup \{y_n:n\in \mathbb{N}\}.$ Since $f$ is a $k$-$cs$-mapping at $A,$ $f^{-1}(T)$ is a compact subset of $X.$

Claim 1. $f$ is a sequentially quotient mapping at $A.$

Indeed, for each $n\in \mathbb{N},$ choose $x_n\in f^{-1}(y_n).$ Since $f^{-1}(T)$ is compact in the metric space $X,$ there exists a subsequence ${\{x_{n_i}\}}_{i\in \mathbb{N}}$ of ${\{x_n\}}_{n\in \mathbb{N}}$ such that ${\{x_{n_i}\}}_{i\in \mathbb{N}}$ converges to some $x\in f^{-1}(T)$ in $X$. This implies that ${\{f(x_{n_i})\}}_{i\in \mathbb{N}}$ is a subsequence of ${\{y_n\}}_{n\in \mathbb{N}}.$ Therefore, $f$ is a sequentially quotient mapping at $A.$

Claim 2. $f$ is a $cs$-regular mapping at $A.$

Assume that $d$ is a metric on $X$ and $U$ is an open neighborhood of $T$ in $Y.$ If $d(f^{-1}(T),X\setminus f^{-1}(U))=0,$ then there exist sequences ${\{x_n\}}_{n\in \mathbb{N}}\subset f^{-1}(T)$ and ${\{z_n\}}_{n\in \mathbb{N}}\subset X\setminus f^{-1}(U)$ such that ${lim}_{n\to \mathrm{\infty }}d(x_n,z_n)=0.$ Since $f^{-1}(T)$ is compact in the metric space $X,$ there exists a subsequence ${\{x_{n_i}\}}_{i\in \mathbb{N}}$ of ${\{x_n\}}_{n\in \mathbb{N}}$ such that ${\{x_{n_i}\}}_{i\in \mathbb{N}}$ converges to some $x\in f^{-1}(T)$ in $X.$ Moreover, since

$$0\le d(z_{n_i},x)\le d(z_{n_i},x_{n_i})+d(x_{n_i},x)\text{for all }i\in \mathbb{N},$$

we have that ${lim}_{i\to \mathrm{\infty }}{z}_{n_i}=x.$ Hence,

$$x\in \overline{X\setminus f^{-1}(U)}=X\setminus f^{-1}(U)\subset X\setminus f^{-1}(T),$$

which contradicts the fact that $x\in f^{-1}(T).$ Therefore,

$$d(f^{-1}(T),X\setminus f^{-1}(U))>0.$$

Thus, $f$ is a $cs$-regular mapping at $A.$

Lemma 3.6 ([7], Lemma 3.9).

Let $𝒫$ be a familiy of subsets of a first countable space $X.$ Then, $𝒫$ is locally finite if and only if it is $cs$-finite.

Lemma 3.7 ([9], Theorem 4.4.7).

A space $X$ is metrizable if and only if $X$ is regular and has a $\sigma$-locally finite base.

Lemma 3.8 ([9], Exercise 5.4.E(a)).

A space $X$ is metrizable if and only if $X$ has a compact-star network consisting of open covers.

Lemma 3.9 ([21], Proposition 3.2).

Suppose that $A$ is a subset of a space $X.$ If $X$ has a point-regular $c{s}^{\ast }$-network at $A$ for $X,$ then $X$ has a sequence of point-countable $c{s}^{\ast }$-covers at $A,$ which is a point-star network at $A$ for $X.$

Lemma 3.10.

Let $X$ be a space. If $U$ is a sequential neighborhood of a point $x$ in $X,$ and ${\{P_n\}}_{n\in \mathbb{N}}$ is a decreasing network at $x$ in $X,$ then $P_m\subset U$ for some $m\in \mathbb{N}.$

Proof 3.11.

Otherwise, then for each $n\in \mathbb{N},$ there exists $x_n\in P_n\setminus U.$ Since ${\{P_n\}}_{n\in \mathbb{N}}$ is a decreasing network at $x$ in $X,$ the sequence ${\{x_n\}}_{n\in \mathbb{N}}$ converges to $x$ in $X.$ This implies that ${\{x_n\}}_{n\in \mathbb{N}}$ is eventually in $U,$ which contradicts that $x_n\notin U$ for every $n\in \mathbb{N}.$

Theorem 3.12.

Let $A$ be a sequentially open subset of a space $X.$ Then, the following are equivalent.

1. $A$ is $cs$-metrizable.

2. $X$ has an $so$-regular $so$-network at $A$ for $X$.

3. $X$ has a $cs$-regular $c{s}^{\ast }$-network at $A$ for $X$.

4. $X$ has a sequence of $so$-finite $so$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

5. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

6. $X$ has a sequence of $c{s}^{\prime }$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

7. $X$ has a sequence of $so$-finite $so$-covers at $A$ which is a point-2-star network at $A$ for $X$.

8. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers at $A$ which is a point-2-star network at $A$ for $X$.

9. $X$ is an image of a metric space under a $k$-$cs$-(and sequence-covering) mapping at $A.$

10. $X$ has a $\sigma$-$cs$-finite $so$-network at $A$ for $X$ and $sA$ is a regular space.

Proof 3.13.

The implications $(2)\Rightarrow (3)$, $(4)\Rightarrow (5)$, and $(7)\Rightarrow (8)$ are immediate. By [27, Theorem 3.11], we have $(6)\Rightarrow (1)$. Moreover, $(4)\Rightarrow (7)$ and $(8)\Rightarrow (5)$ follow from Remark 2.4 and Proposition 3.2. On the other hand, $(9)\Rightarrow (6)$ follows from Lemma 3.4 and [27, Theorem 3.5]. We further prove the remaining implications.

$(1)\Rightarrow (4).$ Assume that $A$ is $cs$-metrizable, i.e., $sA$ is metrizable. Then, $sA$ has a $cs$-star network ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ consisting of locally finite open covers by Lemma 3.8, [9, Theorem 4.4.1] and Remark 2.7(2). Since $A$ is a sequentially open subset of $X$, we claim that ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of $so$-finite $so$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

$(5)\Rightarrow (3).$ Let ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ be a sequence of $cs$-finite $c{s}^{\ast }$-covers at $A$ which is a $cs$-star network at $A$ for $X$. We can assume that ${𝒫}_{n+1}$ refines ${𝒫}_n$ for each $n\in \mathbb{N}.$ Put $𝒫={\bigcup }_{n\in \mathbb{N}}{𝒫}_n.$ Suppose that $x\in A$ and $U$ is an open neighborhood of $x$ in $X$.

Claim 1. $𝒫$ is $cs$-regular at $A$ for $X$.

Let ${\{x_k\}}_{k\in \mathbb{N}}$ be a sequence in $A$ converging to $x.$ For each $n\in \mathbb{N},$ since ${𝒫}_n$ is $cs$-finite, ${({𝒫}_n)}_{T[x_k]}$ is finite. Since ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X$, there exists $m\in \mathbb{N}$ such that $\text{St}(T{[x_k]}_m,{𝒫}_n)\subset U$ for all $n\ge m.$ Then,

This implies that $\{P\in {(𝒫)}_{T{[x_k]}_m}:P\not\subset U\}$ is finite. Therefore, $𝒫$ is $cs$-regular at $A$ for $X$.

Claim 2. $𝒫$ is a $c{s}^{\ast }$-network at $A$ for $X$.

Assume that ${\{x_k\}}_{k\in \mathbb{N}}$ is a sequence converging to $x$ in $X$. Since ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X$, there exists $m\in \mathbb{N}$ such that $\text{St}(x,{𝒫}_m)\subset U.$ Moreover, since ${𝒫}_m$ is a $c{s}^{\ast }$-cover at $A$ for $X$, there exists $P_m\in {𝒫}_m$ such that ${\{x_k\}}_{k\in \mathbb{N}}$ is frequently in $P_m\subset U.$ This shows that $𝒫$ is a $c{s}^{\ast }$-network at $A$ for $X$.

By Claims 1 and 2, (3) holds.

$(3)\Rightarrow (6).$ Let $𝒫$ be a $cs$-regular $c{s}^{\ast }$-network at $A$ for $X$. We can assume that $𝒫$ is closed under finite intersections and $\{\{x\}:x\in S(X)\cap A\}\subset 𝒫.$ Put

${𝒫}_1$ $={𝒫}^m,$

${𝒫}_{n+1}$ $={\left[\left(𝒫\setminus \left({\displaystyle \bigcup _{s=1}^n}{𝒫}_s\right)\right)\cup \{\{x\}:x\in S(X)\cap A\}\right]}^m\text{ for each }n\in \mathbb{N}.$

It is easy to see that each ${𝒫}_{n+1}$ refines ${𝒫}_n$. By Lemma 3.9, $𝒫={\bigcup }_{n\in \mathbb{N}}{𝒫}_n,$ and ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of point-countable $c{s}^{\ast }$-covers at $A$ which is a point-star network at $A$ for $X$. Next, we show that ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a $cs$-star network at $A$ for $X$. Indeed, assume that ${\{x_k\}}_{k\in \mathbb{N}}$ is a sequence converging to $x$ in $X$ and $U$ is an open set in $X$ such that $T[x_k]\subset U\cap A.$ Since $𝒫$ is a $cs$-regular at $A$ for $X$, there exists $i\in \mathbb{N}$ such that ${𝒫}_U=\{P\in {(𝒫)}_{T{[x_k]}_i}:P\not\subset U\}$ is finite. We can assume that ${𝒫}_U\ne \mathrm{\varnothing }.$ For each $P\in {𝒫}_U,$ there exists $n_P\in \mathbb{N}$ such that $P\in {𝒫}_{n_P}.$ Put

$$n_0=1+\max \{n_P:P\in {𝒫}_U\}.$$

Then, $\text{St}(T{[x_k]}_i,{𝒫}_{n_0})\subset U.$ Otherwise, there exists $P_{n_0}\in {𝒫}_{n_0}$ such that $P_{n_0}\cap T{[x_k]}_i\ne \mathrm{\varnothing }$ and $P_{n_0}\not\subset U,$ which implies that $P_{n_0}\in {𝒫}_U.$ Hence,

$$P_{n_0}\in \bigcup _{s=1}^{n_0-1}{𝒫}_s\text{and}{P}_{n_0}\notin \{\{x\}:x\in S(X)\cap A\}.$$

By the construction of the sequence ${\{{𝒫}_n\}}_{n\in \mathbb{N}},$ we have that $P_{n_0}\notin {𝒫}_n$ for all $n\ge n_0.$ This contradicts the fact that $P_{n_0}\in {𝒫}_{n_0}.$ Therefore, $\text{St}(T{[x_k]}_i,{𝒫}_{n_0})\subset U.$

For each since $U$ is an open neighborhood of $x_k$ in $X$ and ${\{\text{St}(x_k,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x_k$ in $X,$ there exists $n_k\in \mathbb{N}$ such that $\text{St}(x_k,{𝒫}_{n_k})\subset U.$ Now, let

Then, we have that

Thus, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of $c{s}^{\prime }$-covers at $A$ which is a $cs$-star network at $A$ for $X$.

$(1)\Rightarrow (2)$ and $(1)\Rightarrow (10).$ Assume that $A$ is $cs$-metrizable, i.e., $sA$ is metrizable. Then, $sA$ is a regular space. Let $d$ be a metric inducing the topology of $sA.$ For each $n\in \mathbb{N},$ there exists a locally finite open refinement ${ℬ}_n$ of the open cover $\{B_d(x,1/n):x\in A\}$ of $sA.$ Put $ℬ={\bigcup }_{n\in \mathbb{N}}{ℬ}_n.$ It is easy to see that $ℬ$ is a $\sigma$-locally finite base of $sA.$ So, $ℬ$ is a $\sigma$-$so$-finite $so$-network of $A.$ Since $A$ is a sequentially open subset of $X,$ $ℬ$ is a $\sigma$-$so$-finite $so$-network at $A$ for $X$.

Now, we show that $ℬ$ is $so$-regular at $A$ for $X$. Assume that $x\in A$ and $U$ is an open neighborhood of $x$ in $X.$ Then, $U\cap A$ is an open neighborhood of $x$ in $sA,$ and hence there exists $m\in \mathbb{N}$ such that $B_d(x,3/m)\subset U\cap A.$ For each $n\le m,$ since ${ℬ}_n$ is locally finite in $sA,$ there exists an open neighborhood $V_n$ of $x$ in $sA$ such that

$$\{P\in {({ℬ}_n)}_{V_n}:P\not\subset U\cap A\}$$

is finite. Put

$$W=B_d(x,1/m)\cap \left(\bigcap _{n=1}^m{V}_n\right).$$

Since $A$ is a sequentially open subset of $X,$ $W$ is sequentially open in $X.$

For each $n>m,$ we have that

$$\text{St}(B_d(x,1/m),{ℬ}_n)\subset U.$$

Indeed, let $y\in \text{St}(B_d(x,1/m),{ℬ}_n).$ Then, there exist $B_n\in {ℬ}_n$ and $z\in A$ such that $y\in B_n$ and $z\in B_n\cap B_d(x,1/m).$ Thus, there exists $t\in A$ such that $B_n\subset B_d(t,1/n).$ It follows that

Hence, $y\in B_d(x,3/m)\subset U$. So, $\text{St}(B_d(x,1/m),{ℬ}_n)\subset U.$

Consequently,

$$\{P\in {(ℬ)}_W:P\not\subset U\}\subset \bigcup _{n=1}^m\{P\in {({ℬ}_n)}_{V_n}:P\not\subset U\cap A\}.$$

Thus, $\{P\in {(ℬ)}_W:P\not\subset U\}$ is finite. It shows that $ℬ$ is $so$-regular at $A$ for $X$.

Therefore, (2) and (10) hold.

$(1)\Rightarrow (9).$ Let $A$ be $cs$-metrizable, i.e, $sA$ is metrizable. Let $d_A$ be a compatible metric on $sA.$ By [9, Theorem 4.1.3], we can assume that $d_A$ is bounded by 1. Suppose that a function $d:X\times X\to [0,\mathrm{\infty })$ is defined as follows:

Then, ${d|}_{A\times A}=d_A$ and $d$ is a metric on $X.$ Hence, $sA$ is a subspace of the metric space $(X,d).$ Moreover, let $\tau$ is a topology on $X$, and let ${\tau }_d$ is the topology induced by the metric $d$. It is easy to check that $\tau \subset {\tau }_d$. Consider the identity mapping ${\text{id}}_X:(X,d)\to (X,\tau )$. We have that ${\text{id}}_X$ is continuous. Next, let $T$ be a convergent set in $A.$ Since $A$ and $sA$ have the same convergent sequences, ${\text{id}}_X^{-1}(T)=T$ is a convergent set in $sA.$ This implies that ${\text{id}}_X^{-1}(T)$ is a convergent set in $(X,d).$ Therefore, ${\text{id}}_X$ is a sequence-covering and $k$-$cs$-mapping at $A$ for $X$. This shows that (9) holds.

$(10)\Rightarrow (1).$ Let $𝒫$ be a $\sigma$-$cs$-finite $so$-network at $A$ for $X$ and $sA$ be a regular space. We can assume that $𝒫$ is closed under finite intersections. Clearly, ${𝒫|}_A$ is a $\sigma$-$cs$-finite $so$-network of $A$. Next, we will prove that ${𝒫|}_A$ is a $\sigma$-$cs$-finite base of $sA.$ Indeed, since $A$ and $sA$ have the same convergent sequences, ${𝒫|}_A$ is a $\sigma$-$cs$-finite family consisting of open sets in $sA.$ Assume that $x\in sA$ and $U$ is an open neighborhood of $x$ in $sA.$ Since ${𝒫|}_A$ is a $\sigma$-$cs$-finite family, ${({𝒫|}_A)}_x$ is countable. Put ${({𝒫|}_A)}_x={\{P_n\}}_{n\in \mathbb{N}}.$ By Lemma 3.10, there exists $m\in \mathbb{N}$ such that $x\in {\bigcap }_{i\le m}{P}_i\subset U.$ On the other hand, since $𝒫$ is closed under finite intersections, ${\bigcap }_{i\le m}{P}_i\in {𝒫|}_A.$ It follows that ${𝒫|}_A$ is a $\sigma$-$cs$-finite base of $sA.$ By Lemma 3.6, ${𝒫|}_A$ is a $\sigma$-locally finite base of $sA.$ On the other hand, since $sA$ is a regular space, by Lemma 3.7, $sA$ is metrizable. Consequently, $A$ is $cs$-metrizable.

Combining the above implications, all conditions $(1)$–$(10)$ are equivalent.

Example 3.14.

There is a subset $A$ of a topological space $X$ satisfying the following conditions:

1. $A$ is $cs$-metrizable;

2. $X$ has not an $so$-regular $so$-network at $A$ for $X$;

3. $X$ has not a sequence of $so$-covers at $A$ which is a $cs$-star network at $A$ for $X.$

Proof 3.15.

In fact, let $X$ be the sequential fan $S_{\omega }$ and $a$ be the unique non-isolated point of $X$. Then, $X$ is not first countable at $a$. Put $A=\{a\}$. Then, $A$ is $cs$-metrizable. Since $X$ is a sequential space, every sequentially open subset of $X$ is open. If $X$ has an $so$-regular $so$-network at $A$ for $X$, it is easy to see that $X$ has a countable local base at $a$, which is a contradiction. If $X$ has a sequence of $so$-covers at $A$ which is a $cs$-star network at $A$ for $X$, then $X$ has a countable local base at $a$, which is a contradiction.

Remark 3.16.

Example 3.14 implies that Theorem 3.12 does not hold without the assumption that $A$ is a sequentially open subset of $X$. Moreover, it provides a negative answer to [27, Question 3.13] for [27, Theorem 3.11].

Lemma 3.17 ([23], Lemma 3.7(2)).

Let $𝒫$ be a family of subsets of a first countable space $X.$ If $𝒫$ is $cs$-regular, then it is regular.

Lemma 3.18 ([9], Lemma 5.4.5).

Let $ℬ$ be a point-regular (regular) base for a space $X$, then letting

${ℬ}_1$ $={ℬ}^m,$

${ℬ}_{n+1}$ $={\left[\left(ℬ\setminus \left({\displaystyle \bigcup _{s=1}^n}{ℬ}_s\right)\right)\cup \{\{x\}:x\in I(X)\}\right]}^m\text{ for every }n\in \mathbb{N},$

we define a sequence ${\{{ℬ}_n\}}_{n\in \mathbb{N}}$ of point-finite (locally finite) open covers of $X$ such that $ℬ={\bigcup }_{n\in \mathbb{N}}{ℬ}_n$.

Theorem 3.19.

Let $A$ be a sequentially open subset of a space $X.$ Then, the following are equivalent.

1. $X$ has a regular $so$-network at $A$ for $X$.

2. $X$ has a regular $cs$-network at $A$ for $X$.

3. $X$ has a sequence of regular and $so$-finite $so$-covers at $A$ which is an $o$-star network at $A$ for $X$.

4. $X$ has a sequence of regular $cs$-covers at $A$ which is an $o$-star network at $A$ for $X$.

Proof 3.20.

$(2)\Rightarrow (1).$ Suppose that $𝒫$ is a regular $cs$-network at $A$ for $X$. We can assume that $𝒫$ is closed under finite intersections and that $\{\{x\}:x\in S(X)\cap A\}\subset 𝒫$. Put

$$𝒬=\{{\text{Int}}_{sX}(P):P\in 𝒫\}.$$

Now, we will prove that $𝒬$ is a regular $so$-network at $A$ for $X.$ Let $x\in A$ and let $U$ be an open neighborhood of $x$ in $X.$ By Theorem 3.12, $X$ has a $\sigma$-$cs$-finite $so$-network $𝒢$ at $A$ for $X$. Then, ${(𝒢)}_x=\{G_n:n\in \mathbb{N}\}$ is a countable network at $x$ consisting of sequentially open sets in $X.$ Since $𝒫$ is a regular $cs$-network at $A$ for $X$, by Lemma 3.9, $𝒫$ is point-countable at $A$ for $X$. Put

$${𝒫}^{\prime }=\{P\in {(𝒫)}_x:P\subset U\}={\{P_k\}}_{k\in \mathbb{N}}.$$

Then, there exist $n,k\in \mathbb{N}$ such that ${\bigcap }_{i=1}^n{G}_i\subset P_k.$ Otherwise, for each $n\in \mathbb{N}$ and each $k\in \mathbb{N},$ there exists

$$x_{n,k}\in \left(\bigcap _{i=1}^n{G}_i\right)\setminus P_k.$$

For each $n,k\in \mathbb{N}$ with $n\ge k,$ put $m=k+n(n-1)/2$ and $y_m=x_{n,k}.$ Since ${\{G_n\}}_{n\in \mathbb{N}}$ is a network at $x$ in $X,$ the sequence ${\{y_m\}}_{m\in \mathbb{N}}$ converges to $x$ in $X.$ On the other hand, since $𝒫$ is a $cs$-network at $A$ for $X$, there exist $r,s\in \mathbb{N}$ such that $\{y_m:m\ge r\}\subset P_s.$ Choose $j\ge s$ such that $l=s+j(j-1)/2\ge r.$ Then, $x_{j,s}=y_l\in P_s,$ which contradicts the fact that $x_{n,k}\notin P_k$ for all $n,k\in \mathbb{N}.$ Therefore, there exist $n,k\in \mathbb{N}$ such that ${\bigcap }_{i=1}^n{G}_i\subset P_k.$ Consequently, we claim that

$$x\in \bigcap _{i=1}^n{G}_i={\text{Int}}_{sX}\left(\bigcap _{i=1}^n{G}_i\right)\subset {\text{Int}}_{sX}(P_k)\subset U.$$

This shows that $𝒬$ is an $so$-network at $A$ for $X.$ Moreover, it is easy to see that $𝒬$ is regular at $A$ for $X$. Hence, $𝒬$ is a regular $so$-network at $A$ for $X$.

$(1)\Rightarrow (3).$ Let $𝒫$ be a regular $so$-network at $A$ for $X$. It follows from Theorem 3.12 that $A$ is $cs$-metrizable. We can assume that $𝒫$ is closed under finite intersections and $\{\{x\}:x\in S(X)\cap A\}\subset 𝒫.$ By Lemma 3.9, $𝒫$ is point-countable at $A$ for $X$. Now, we will show that ${𝒫|}_A$ is a $cs$-regular base of $sA.$ Let $x\in A,$ let $U$ be an open neighborhood of $x$ in $sA,$ and let ${\{x_n\}}_{n\in \mathbb{N}}$ be a sequence converging to $x$ in $sA.$ Put ${({𝒫|}_A)}_x={\{P_j\}}_{j\in \mathbb{N}}.$ Clearly, each element of ${𝒫|}_A$ is open in $sA.$

Claim 1. There exists $j_0\in \mathbb{N}$ such that ${\bigcap }_{s=1}^{j_0}{P}_s\subset U$. This follows directly from Lemma 3.10.

Claim 2. ${𝒫|}_A$ is $cs$-regular in $sA.$

Otherwise, $\{P\in {({𝒫|}_A)}_{T{[x_n]}_i}:P\not\subset U\}$ is infinite for every $i\in \mathbb{N}.$ Hence, there exists a sequence ${\{F_i\}}_{i\in \mathbb{N}}$ of distinct elements of ${𝒫|}_A$ such that $F_i\cap T{[x_n]}_i\ne \mathrm{\varnothing }$ and $F_i\not\subset U$ for each $i\in \mathbb{N}.$ Thus, for each $i\in \mathbb{N},$ there exists $z_i\in F_i\setminus U.$ We claim that the sequence ${\{z_i\}}_{i\in \mathbb{N}}$ converges to $x$ in $X.$ Indeed, let $V$ be an open neighborhood of $x$ in $X.$ Since $𝒫$ is $cs$-regular at $A$ for $X$, there exists $m\in \mathbb{N}$ such that

$$𝒬=\{H\in {(𝒫)}_{T{[x_n]}_m}:H\not\subset V\}$$

is finite, and hence ${𝒬|}_A$ is finite. Moreover, since $F_i\cap T{[x_n]}_m\ne \mathrm{\varnothing }$ for all $i\ge m$ and

$$\{P\in {(𝒫{|}_A)}_{T{[x_n]}_m}:P\not\subset V\}\subset 𝒬{|}_A,$$

there exists $k\in \mathbb{N}$ such that $F_i\subset V$ for all $i\ge k.$ Thus, the sequence ${\{z_i\}}_{i\in \mathbb{N}}$ converges to $x$ in $X.$ Furthermore, since $A$ is sequentially open in $X,$ $U$ is sequentially open in $X,$ and hence the sequence ${\{z_i\}}_{i\in \mathbb{N}}$ is eventually in $U.$ This contradicts the fact that $z_i\in F_i\setminus U$ for all $i\in \mathbb{N}.$ Therefore, ${𝒫|}_A$ is $cs$-regular in $sA.$

Since $𝒫$ is closed under finite intersections, ${\bigcap }_{s=1}^{j_0}{P}_s\in {({𝒫|}_A)}_x.$ By Claims 1 and 2, we claim that ${𝒫|}_A$ is a $cs$-regular base of $sA.$

Next, since $sA$ is first countable, ${𝒫|}_A$ is a regular base of $sA$ by Lemma 3.17. Put

${𝒫}_1$ $={({𝒫|}_A)}^m,$

${𝒫}_{n+1}$ $={\left[\left({𝒫|}_A\setminus \left({\displaystyle \bigcup _{s=1}^n}{𝒫}_s\right)\right)\cup \{\{x\}:x\in I(sA)\}\right]}^m\text{for all }n\in \mathbb{N}.$

It follows from Lemma 3.18 that each ${𝒫}_n$ is a locally finite open cover of $sA,$ and ${𝒫|}_A={\bigcup }_{n\in \mathbb{N}}{𝒫}_n.$ Hence, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of $so$-finite $so$-covers of $A.$ Since $A$ is sequentially open in $X,$ ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of $so$-finite $so$-covers at $A$ for $X$. Since $𝒫$ is regular at $A$ for $X$, it is easy to check that ${𝒫|}_A$ is regular at $A$ for $X$. Moreover, since each ${𝒫}_n$ is a subfamily of ${𝒫|}_A,$ it is regular at $A$ for $X$.

Finally, let $x\in A$ and let $U$ be an open neighborhood of $x$ in $X.$ Since ${𝒫|}_A$ is regular at $A$ for $X$, there exists an open neighborhood $V$ of $x$ in $X$ such that $V\subset U$ and $\{P\in {({𝒫|}_A)}_V:P\not\subset U\}$ is finite. By the construction of the sequence ${\{{𝒫}_n\}}_{n\in \mathbb{N}},$ there exists $n_V\in \mathbb{N}$ such that $\text{St}(V,{𝒫}_{n_V})\subset U.$ Therefore, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is an $o$-star network at $A$ for $X$. Consequently, ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of regular and $so$-finite $so$-covers at $A$ which is an $o$-star network at $A$ for $X.$

$(3)\Rightarrow (4).$ This is obvious.

$(4)\Rightarrow (2).$ Suppose that ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is a sequence of regular $cs$-covers at $A$ which is an $o$-star network at $A$ for $X.$ We can assume that each ${𝒫}_{n+1}$ refines ${𝒫}_n$. Put $𝒫={\bigcup }_{n\in \mathbb{N}}{𝒫}_n.$ Let $x\in A,$ and let $U$ be an open neighborhood of $x$ in $X.$

Claim 1. $𝒫$ is regular at $A$ for $X$.

For each $n\in \mathbb{N},$ since ${𝒫}_n$ is regular at $A$ for $X$, there exists an open neighborhood $V_n$ of $x$ in $X$ such that $\{P\in {({𝒫}_n)}_{V_n}:P\not\subset U\}$ is finite. Since ${\{{𝒫}_n\}}_{n\in \mathbb{N}}$ is an $o$-star network at $A$ for $X$, there exist an open neighborhood $V$ of $x$ in $X$ and $m\in \mathbb{N}$ such that $\text{St}(V,{𝒫}_n)\subset U$ for all $n>m.$ Put $W=\left({\bigcap }_{i=1}^m{V}_i\right)\cap V.$ Then, $W$ is an open neighborhood of $x$ in $X$ and

$$\{P\in {(𝒫)}_W:P\not\subset U\}\subset \bigcup _{i=1}^m\{P\in {({𝒫}_i)}_W:P\not\subset U\},$$

which implies that $\{P\in {(𝒫)}_W:P\not\subset U\}$ is finite. Therefore, $𝒫$ is regular at $A$ for $X$.

Claim 2. $𝒫$ is a $cs$-network at $A$ for $X$.

Let $x\in A$ and let ${\{x_n\}}_{n\in \mathbb{N}}$ be a sequence converging to $x\in U$ in $X,$ where $U$ is open in $X.$ Since ${\{\text{St}(x,{𝒫}_n)\}}_{n\in \mathbb{N}}$ is a network at $x$ in $X,$ there exists $m\in \mathbb{N}$ such that $\text{St}(x,{𝒫}_m)\subset U.$ Moreover, since ${𝒫}_m$ is a $cs$-cover at $A$ for $X$, there exists $P_m\in {𝒫}_m$ such that ${\{x_n\}}_{n\in \mathbb{N}}$ is eventually in $P_m\subset U.$ Therefore, $𝒫$ is a $cs$-network at $A$ for $X$.

By Claims 1 and 2, $X$ has a regular $cs$-network at $A$ for $X$.

Question 3.21.

Let $A\subset X$. If $X$ has a regular $c{s}^{\ast }$-network at $A$, does $X$ have a regular $cs$-network at $A$?

Question 3.22.

Is Theorem 3.19 still true if $A$ is not a sequentially open subset of $X$?

In Theorems 3.12 and 3.19, if $A=X,$ then we obtain the following corollaries, which are results previously given by S. Lin, W. Zheng and Z.Y. Cai in [23].

Corollary 3.23 ([23], Theorems 4.2, 4.6).

The following are equivalent for a space $X.$

1. $X$ is $cs$-metrizable.

2. $X$ has an $so$-regular $so$-network.

3. $X$ has a $cs$-regular $c{s}^{\ast }$-network.

4. $X$ has a sequence of $so$-finite $so$-covers which is a $cs$-star network.

5. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers which is a $cs$-star network.

6. $X$ has a sequence of $so$-finite $so$-covers which is a point-2-star network.

7. $X$ has a sequence of $cs$-finite $c{s}^{\ast }$-covers which is a point-2-star network.

8. $X$ is a $k$-$cs$-(and sequence-covering) image of a metric space.

9. $X$ has a $\sigma$-$cs$-finite $so$-network and $sX$ is a regular space.

Corollary 3.24 ([23], Theorem 5.2).

The following are equivalent for a space $X$.

1. $X$ has a regular $so$-network.

2. $X$ has a regular $cs$-network.

3. $X$ has a sequence of regular and $so$-finite $so$-covers which is an $o$-star network.

4. $X$ has a sequence of regular $cs$-covers which is an $o$-star network.