@
Appl. Gen. Topol. 21, no. 2 (2020), 235-246

doi:10.4995/agt.2020.12967

c© AGT, UPV, 2020

On I-quotient mappings and I-cs′-networks
under a maximal ideal

Xiangeng Zhou∗

Department of Mathematics, Ningde Normal University, Fujian, 352100, P.R. China (56667400@qq.com)

Communicated by P. Das

Abstract

Let I be an ideal on N and f : X → Y be a mapping. f is said to
be an I-quotient mapping provided f−1(U) is I-open in X, then U is
I-open in Y . P is called an I-cs′-network of X if whenever {xn}n∈N is
a sequence I-converging to a point x ∈ U with U open in X, then there
is P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U. In this paper,
we introduce the concepts of I-quotient mappings and I-cs′-networks,
and study some characterizations of I-quotient mappings and I-cs′-
networks, especially J -quotient mappings and J -cs′-networks under
a maximal ideal J of N. With those concepts, we obtain that if X
is an J -FU space with a point-countable J -cs′-network, then X is a
meta-Lindelöf space.

2010 MSC: 54A20; 54B15; 54C08; 54D55; 40A05; 26A03.

Keywords: ideal convergence; maximal ideal; I-sequential neighborhood;
I-quotient mappings; I-cs′-networks; I-FU spaces.

1. Introduction

Statistical convergence was introduced by H. Fast [9] and H. Steinhaus [16],
which is a generalization of the usual notion of convergence. It is doubtless that
the study of statistical convergence and its various generalizations has become
an active research area [2, 3, 7, 17, 18]. In particular, P. Kostyrko, T. Šalát

∗This research is supported by NSFC (No. 11801254) and Ningde Normal University (No.
2017T01; 2018ZDK11; 2019ZDK11).

Received 10 January 2020 – Accepted 11 April 2020

http://dx.doi.org/10.4995/agt.2020.12967


X. Zhou

and W. Wilczynski [11] introduced two interesting generalizations of statistical
convergence by using the notion of ideals of subsets of positive integers, which
were named as I and I∗-convergence, and studied some properties of I and
I∗-convergence in metric spaces. Later, B.K. Lahiri and P. Das [12] discussed
I and I∗-convergence in topological spaces. Some further results connected
with I and I∗-convergence can be found in [4, 5, 6].

As we know, mappings and networks are important tools of investigating
topological spaces. Continuous mappings, quotient mappings , pseudo-open
mappings, cs-networks, sn-networks, k-networks and so on are the most impor-
tant tools for studying convergence, sequential spaces, Fréchet-Urysohn spaces
[14] and generalized metric spaces. For this reason, this paper draws into I-
quotient mappings and I-cs′-networks for an ideal I on N and discusses some
basic properties of them.

Recently, the researches on I-convergence are mainly focused on aspects of
I∗-convergence [12], I-limit points [11], I-Cauchy sequence [5], ideal-convergence
classes [4], selection principles [6], ideal sequence covering mappings [15, 19] and
so on. It is expected that I-quotient mappings and I-cs′-networks will also play
active roles in the topological spaces.

In this paper, the letter X always denote a topological space. The cardi-
nality of a set B is denoted by |B|. The set of all positive integers, the first
infinite ordinal, and the first uncountable ordinal are denoted by N, ω and ω1,
respectively. The reader may refer to [8, 14] for notation and terminology not
explicitly given here.

2. Preliminaries

Recall the notion of statistical convergence in topological spaces. For each
subset A of N the asymptotic density of A, denoted δ(A), is given by

δ(A) = lim
n→∞

1

n
|{k ∈ A : k ≤ n}|,

if this limit exists. Let X be a topological space. A sequence {xn}n∈N in X is
said to converge statistically to a point x ∈ X [7], if

δ({n ∈ N : xn ∕∈ U}) = 0, i.e., δ({n ∈ N : xn ∈ U}) = 1
for each neighborhood U of x in X, which is denoted by s- lim

n→∞
xn = x or

xn
s−→ x.
The concept of I-convergence of sequences in a topological space is a gen-

eralization of statistical convergence which is based on the ideal of subsets of
the set N of all positive integers. Let A = 2N be the family of all subsets of
N. An ideal I ⊆ A is a hereditary family of subsets of N which is stable under
finite unions [11], i.e., the following are satisfied: if B ⊆ A ∈ I, then B ∈ I; if
A, B ∈ I, then A ∪ B ∈ I. An ideal I is said to be non-trivial, if I ∕= ∅ and
N /∈ I. A non-trivial ideal I ⊆ A is called admissible if I ⊇ {{n} : n ∈ N}.
Clearly, every non-trivial ideal I defines a dual filter FI = {A ⊆ N : N\A ∈ I}
on N.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 236



On I-quotient mappings and I-cs′-networks under a maximal ideal

Let If be the family of all finite subsets of N. Then If is an admissible
ideal. Let Iδ [11] be the family of subsets A ⊆ N with δ(A) = 0. Then Iδ is an
admissible ideal, and the dual filter FIδ = {A ⊆ N : δ(A) = 1}.

Definition 2.1 ([11]). A sequence {xn}n∈N in a topological space X is said to
be I-convergent to a point x ∈ X provided for any neighborhood U of x, we
have {n ∈ N : xn /∈ U} ∈ I, which is denoted by I- lim

n→∞
xn = x or xn

I−→ x,
and the point x is called the I-limit of the sequence {xn}n∈N.

Definition 2.2 ([20]). Let I be an ideal on N and X be a topological space.
(1) A subset F ⊆ X is said to be I-closed if for each sequence {xn}n∈N ⊆ F

with xn
I−→ x ∈ X, we have x ∈ F .

(2) A subset U ⊆ X is said to be I-open if X \ U is I-closed.
(3) X is called an I-sequential space if each I-closed subset of X is closed.

Obviously, each sequential space is an I-sequential space [20].

Definition 2.3 ([20]). Let I be an ideal on N, X, Y be topological spaces and
f : X → Y be a mapping.

(1) f is called preserving I-convergence provided for each sequence {xn}n∈N
in X with xn

I−→ x, the sequence {f(xn)}n∈N I-converges to f(x) [12].
(2) f is called I-continuous provided U is I-open in Y , then f−1(U) is

I-open in X.

It is easy to verify that a mapping f : X → Y is I-continuous if and only if,
whenever F is I-closed in Y , then f−1(F) is I-closed in X.

Lemma 2.4 ([20]). Let I be an ideal on N and X be a topological space. If
a sequence {xn}n∈N I-converges to a point x ∈ X, and {yn}n∈N is a sequence
in X with {n ∈ N : xn ∕= yn} ∈ I, then the sequence {yn}n∈N I-converges to
x ∈ X.

Lemma 2.5 ([20]). Let I be an ideal on N. The following are equivalent for a
topological space X and a subset A ⊆ X.

(1) A is I-open.
(2) {n ∈ N : xn ∈ A} /∈ I for each sequence {xn}n∈N in X with xn

I−→ x ∈
A.

(3) |{n ∈ N : xn ∈ A}| = ω for each sequence {xn}n∈N in X with xn
I−→

x ∈ A.

Lemma 2.6 ([20]). Let X, Y be topological spaces and f : X → Y be a
mapping.

(1) If f is continuous, then f preserves I-convergence [12].
(2) If f preserves I-convergence, then f is I-continuous.

Definition 2.7 ([20]). Let A ⊆ X and {xn}n∈N be a sequence in X. If I is
an ideal on N, then {xn}n∈N is I-eventually in A if there is E ∈ I such that
for all n ∈ N \ E, xn ∈ A.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 237



X. Zhou

If A is a subset of X with the property that every sequence I-converging to
a point in A is I-eventually in A, then A is I-open. When we assume J to be
a maximal ideal, the following proposition shows that such sets must coincide
with J -open sets.

Proposition 2.8 ([20]). If J is a maximal ideal of N, then A ⊆ X is J -open
if and only if for each J -converging sequence {xn}n∈N with xn

J−→ x ∈ A, then
{xn}n∈N is J -eventually in A.

By Definition 2.2, the union of a family of I-open sets in a topological space
is I-open. Whenever J is a maximal ideal, the intersection of two J -open sets
is an J -open set.

Proposition 2.9 ([20]). If J is a maximal ideal of N and U, V are two J -open
subsets of X, then U ∩ V is J -open in X.

It is well known that the sequential coreflection sX of a space X is the set
X endowed with the topology consisting of sequentially open subsets of X. Let
J be a maximal ideal of N and X be a topological space. By Definition 2.2
and Proposition 2.9, the family of all J -open subsets of X forms a topology of
the set X. The J -sequential coreflection of a space X is the set X endowed
with the topology consisting of J -open subsets of X, which is denoted by J -
sX. The spaces X and J -sX have the same J -convergent sequences; J -sX
is an J -sequential space; a space X is an J -sequential space if and only if
J -sX = X [20].

If no otherwise specified, we consider ideal I is always an admissible ideal
on N, all mappings are continuous and surjection, all spaces are Hausdorff.

3. I-quotient mappings

In this section, we introduce the concept of I-quotient mappings, and obtain
some characterizations of I-quotient mappings, especially J -quotient mappings
under a maximal ideal of N. Let X, Y be arbitrary topological spaces, and
f : X → Y be a mapping. f is said to be quotient provided f−1(U) is open in
X, then U is open in Y ; f is said to be sequentially quotient provided f−1(U)
is sequentially open in X, then U is sequentially open in Y [1].

Definition 3.1. Let I be an ideal on N and f : X → Y be a mapping.
(1) f is said to be an I-quotient mapping (or shortly, I-quotient) provided

f−1(U) is I-open in X, then U is I-open in Y .
(2) f is said to be an I-covering mapping (or shortly, I-covering) if, when-

ever {yn}n∈N is a sequence in Y I-converging to y in Y , there exist a
sequence {xn}n∈N of points xn ∈ f−1(yn) for all n ∈ N and x ∈ f−1(y)
such that xn

I−→ x.

In [20], it was showed that each I-covering mapping is I-quotient.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 238



On I-quotient mappings and I-cs′-networks under a maximal ideal

Definition 3.2. Let I be an ideal on N, X be a topological space and P ⊂ X.
P is called an I-sequential neighborhood of x, if each sequence {xn}n∈N I-
converges to a point x ∈ P , then {xn}n∈N is I-eventually in P , i.e., there is
I ∈ I such that {n ∈ N : xn /∈ P} = I.

Remark 3.3. Let J be a maximal ideal of N and A ⊆ X. By Proposition 2.8,
A is J -open in X if and only if A is an J -sequential neighborhood of x for
each x ∈ A.

Proposition 3.4. Let J be a maximal ideal of N and A ⊆ X. If A is not
an J -sequential neighborhood of x, then there is a sequence {xn}n∈N in X \ A
such that xn

J−→ x.

Proof. If A is not an J -sequential neighborhood of x, then there is a sequence
{yn}n∈N in X such that yn

J−→ x, but {n ∈ N : yn /∈ A} /∈ J . Since J is a
maximal ideal of N, this means that {n ∈ N : yn ∈ A} ∈ J . Let {n ∈ N : yn ∈
A} = J ∈ J . And since J is a non-trivial ideal, it follows that A ∕= X. Take
a point a ∈ X \ A. Define a sequence {xn}n∈N by xn = a if n ∈ J; xn = yn
if n ∈ N \ J. Then the sequence {xn}n∈N in X \ A and xn

J−→ x from Lemma
2.5. □

Theorem 3.5. Let I be an ideal on N. If f : X → Y is an I-quotient
mapping, then for each I-convergent sequence {yn}n∈N in Y with yn

I−→ y,
there is a sequence {xi}i∈N in X such that {xi : i ∈ N} ⊆ f−1({yn : n ∈ N})
and xi

I−→ x /∈ f−1({yn : n ∈ N}).

Proof. Suppose that f : X → Y is an I-quotient mapping and {yn}n∈N is a
sequence in Y with yn

I−→ y. Without loss of generality, we can assume that
yn ∕= y for each n ∈ N. Let U = Y \ {yn : n ∈ N}. Then U is not I-open
in Y . Since f is an I-quotient mapping, f−1(U) = f−1(Y \ {yn : n ∈ N}) =
X \ f−1({yn : n ∈ N}) is not I-open in X. Thus there is a sequence {xi}i∈N in
X \ f−1(U) = f−1({yn : n ∈ N}) such that xi

I−→ x /∈ f−1({yn : n ∈ N}). □

In [20], it was discussed that quotient mappings, sequentially quotient map-
pings and I-quotient mappings are mutually independent; and the following
two theorems are useful and can be seen in it.

Theorem 3.6. Let f : X → Y be a mapping.
(1) If X is an I-sequential space and f is quotient, then Y is an I-

sequential space and f is I-quotient.
(2) If Y is an I-sequential space and f is I-quotient, then f is quotient.
(3) X is an I-sequential space if and only if for an arbitrary topological

space Y , if f is quotient, then f is I-quotient.

Theorem 3.7. Let J be a maximal ideal of N and X be a topological space.
Then X is an J -sequential space if and only if each J -quotient mapping onto
X is quotient.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 239



X. Zhou

Let J be a maximal ideal of N and A ⊆ X. Denote
[A]J -s = {x ∈ X : there is a sequence {xn}n∈N in A such that xn

J−→ x};
(A)J -s = {x ∈ X : A is an J -sequential neighborhood of x}.
A subset U ⊆ X is said to be an J -sequential neighborhood of A if A ⊆

(U)J -s.

Proposition 3.8. Let J be a maximal ideal of N and A ⊆ X. Then [A]J -s =
X \ (X \ A)J -s.

Proof. Suppose that x ∈ [A]J -s, then there is a sequence {xn}n∈N in A such
that xn

J−→ x. Thus X \ A is not an J -sequential neighborhood of x in X. In
fact, if X \ A is an J -sequential neighborhood of x in X, then {xn}n∈N is J -
eventually in X \A, i.e., there is E ∈ J such that for all n ∈ N\E, xn ∈ X \A.
Since J is an admissible ideal, this contradicts to {xn}n∈N in A. Therefore
x /∈ (X \ A)J -s, and further x ∈ X \ (X \ A)J -s.

On the other hand, assume that x ∈ X \ (X \ A)J -s, then x /∈ (X \ A)J -s,
and hence X\A is not an J -sequential neighborhood of x in X. By Proposition
3.4, there is a sequence {xn}n∈N in A such that xn

J−→ x. Thus x ∈ [A]J -s. □

By Definition 2.2 and Proposition 3.8, the following proposition is correct.

Proposition 3.9. Let J be a maximal ideal of N and A, B ⊆ X. Then
(1) [∅]J -s = ∅, A◦ ⊆ (A)J -s ⊆ A ⊆ [A]J -s ⊆ A.
(2) A is J -open in X if and only if A = (A)J -s.
(3) A is J -closed in X if and only if A = [A]J -s.
(4) If B ⊆ A, then (B)J -s ⊆ (A)J -s and [B]J -s ⊆ [A]J -s.
(5) (A ∩ B)J -s = (A)J -s ∩ (B)J -s and [A ∪ B]J -s = [A]J -s ∪ [B]J -s.

Proof. We only prove that (5) is true. Since A ∩ B ⊆ A, A ∩ B ⊆ B, it
follows that (A ∩ B)J -s ⊆ (A)J -s, (A ∩ B)J -s ⊆ (B)J -s. Hence (A ∩ B)J -s ⊆
(A)J -s ∩ (B)J -s. On the other hand, assume that x ∈ (A)J -s ∩ (B)J -s. Then
for each sequence {xn}n∈N in X with xn

J−→ x, there is E, F ∈ J , such that for
each n ∈ N \ E, xn ∈ A and for each n ∈ N \ F , xn ∈ B. Since E ∪ F ∈ J and
for each n ∈ N\(E ∪F), xn ∈ A∩B. This means that A∩B is an J -sequential
neighborhood of x in X. Thus x ∈ (A ∩ B)J -s.

Now replace X \ A with A and X \ B with B, it follows that ((X \ A) ∩ (X \
B))J -s = (X \A)J -s ∩(X \B)J -s. Hence [A∪B]J -s = X \(X \(A∪B))J -s =
X \ ((X \ A) ∩ (X \ B))J -s = X \ ((X \ A))J -s ∩ (X \ B))J -s) = (X \ (X \
A)J -s) ∪ (X \ (X \ B)J -s) = [A]J -s ∪ [B]J -s. □

Theorem 3.10. Let J be a maximal ideal of N and f : X → Y be a mapping.
Then the following conditions are equivalent.

(1) For each J -convergent sequence {yn}n∈N in Y with yn
J−→ y, there is

a sequence {xi}i∈N in X with xi
J−→ x ∈ f−1(y) and {xi : i ∈ N} ⊆

f−1({yn : n ∈ N}).
(2) For each A ⊆ Y , it has f([f−1(A)]J -s) = [A]J -s.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 240



On I-quotient mappings and I-cs′-networks under a maximal ideal

(3) If y ∈ [A]J -s ⊆ Y , then f−1(y) ∩ [f−1(A)]J -s ∕= ∅.
(4) If y ∈ [A]J -s ⊆ Y , then there is a point x ∈ f−1(y) such that whenever

V is an J -sequential neighborhood of x, y ∈ [f(V ) ∩ A]J -s.
(5) If y ∈ [A]J -s ⊆ Y , then there is a point x ∈ f−1(y) such that whenever

V is an J -sequential neighborhood of x, f(V ) ∩ A ∕= ∅.
(6) For each y ∈ Y , if U is an J -sequential neighborhood of f−1(y), then

f(U) is an J -sequential neighborhood of y.

Proof. (1) ⇒ (2) Suppose that x ∈ [f−1(A)]J -s. Then there is a sequence
{xn}n∈N in f−1(A) such that xn

J−→ x. Hence {f(xn) : n ∈ N} ⊆ A and
f(xn)

J−→ f(x). This means that f(x) ∈ [A]J -s. Hence f([f−1(A)]J -s) ⊆
[A]J -s.

On the other hand, assume that y ∈ [A]J -s. Then there is a sequence
{yn}n∈N in A such that yn

J−→ y. By the condition (1), there is a sequence
{xi}i∈N in X with {xi : i ∈ N} ⊆ f−1({yn : n ∈ N}) ⊆ f−1(A) and xi

J−→ x ∈
f−1(y). Thus x ∈ [f−1(A)]J -s, hence y = f(x) ∈ f([f−1(A)]J -s), and further
[A]J -s ⊆ f([f−1(A)]J -s).

(2) ⇒ (3) Let y ∈ [A]J -s for each A ⊆ Y . By the condition (2), it follows
that y ∈ f([f−1(A)]J -s). Thus f−1(y) ∩ [f−1(A)]J -s ∕= ∅.

(3) ⇒ (4) Let y ∈ [A]J -s ⊆ Y . By the condition (3), assume that x ∈
f−1(y) ∩ [f−1(A)]J -s. Then there is a sequence {xn}n∈N in f−1(A) such that
xn

J−→ x. If V is an J -sequential neighborhood of x, then there is E ∈ J such
that xn ∈ V for all n ∈ N \ E. Hence f(xn) ∈ f(V ) ∩ A for all n ∈ N \ E and
f(xn)

J−→ f(x). Take a point a ∈ f(V ) ∩ A. Define a sequence {yn}n∈N by
yn = f(xn) if n ∈ N \ E; yn = a if n ∈ E. Then {yn : n ∈ N} ⊆ f(V ) ∩ A and
yn

J−→ f(x) = y from Lemma 2.4. Thus y ∈ [f(V ) ∩ A]J -s.
(4) ⇒ (5) It is clear.
(5) ⇒ (6) Let y ∈ Y and U be an J -sequential neighborhood of f−1(y).

If f(U) is not an J -sequential neighborhood of y, then y ∈ Y \ (f(U))J -s =
[Y \ f(U)]J -s. By the condition (5), it follows that f(U) ∩ (Y \ f(U)) = ∅, a
contradiction.

(6) ⇒ (3) Let y ∈ [A]J -s ⊆ Y . Suppose that f−1(y) ∩ [f−1(A)]J -s =
∅. Then f−1(y) ⊆ X \ [f−1(A)]J -s = (X \ f−1(A))J -s. This means that
X \ f−1(A) is an J -sequential neighborhood of f−1(y). By the condition (6),
y ∈ (f(X \ f−1(A)))J -s = (Y \ A)J -s = Y \ [A]J -s, a contradiction.

(3) ⇒ (1) Let {yn}n∈N be an J -convergent sequence in Y with yn
J−→ y.

Put A = {yn : n ∈ N}, then y ∈ [A]J -s. By the condition (3), there is
x ∈ f−1(y) ∩ [f−1(A)]J -s. Hence there is a sequence {xi}i∈N in X with {xi :
i ∈ N} ⊆ f−1(A) ⊆ f−1({yn : n ∈ N}) and xi

J−→ x ∈ f−1(y). □

Remark 3.11.

(1) Theorem 3.5 is different from Lemma 3.10 (1). In Lemma 3.10 (1),

xi
J−→ x ∈ f−1(y). But we don’t know whether the I-limit point x in

f−1(y) or not in Theorem 3.5.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 241



X. Zhou

(2) One of the above six conditions can deduce that f is an J -quotient
mapping.

In fact, let U be non-I-closed in Y . Then there is a sequence {yn}n∈N
in U J -converging to y ∈ Y \U. Thus y ∕= yn for each n ∈ N. By the
assumption of the condition (1), there is a sequence {xi}i∈N in X such
that {xi : i ∈ N} ⊆ f−1({yn : n ∈ N}) ⊆ f−1(U) and xi

J−→ x ∈
f−1(y) /∈ f−1(U). This implies that f−1(U) is non-J -closed in X.
Hence, f is an J -quotient mapping.

(3) If the maximal ideal J is replaced by If in Theorem3.10, then (1) ⇔
(2) ⇔ (3) ⇔ (4) ⇔ (5) ⇔ (6) ⇔ f is an If -quotient mapping. But
the following example shows that there exist a T1 space X, an ideal
I of N and an I-quotient mapping f such that f does not satisfy the
condition (6) of Theorem 3.10.

Example 3.12. There exist a T1 space X, an ideal I of N and an I-quotient
mapping f, but f does not satisfy the condition (6) of Theorem 3.10.

Proof. Let I = {A ⊆ N : A contains at most only finite odd positive integers}.
Then I is an admissible ideal of N . Let Y be the set ω which is endowed with
the finite complement topology. Then Y is a first-countable T1-space. Put
X0 = Y \ {0} and X1 = {2k : k ∈ ω} as the subspaces of the space Y , and
X = X0

!
X1. A mapping f : X → Y is defined by the natural mapping.

It is easy to see that the mapping f is a continuous quotient mapping. Since
X0 and X1 are first-countable space, X is a first-countable space. Thus, X
is an I-sequential space. By Theorem 3.6, it follows that f is an I-quotient
mapping.

Note that the set X1 is open in X and f
−1(0) ⊆ X1, and hence X1 is an

I-sequential neighborhood of f−1(0). For each open neighborhood U of 0 in
Y , {n ∈ N : n /∈ U} is a finite subset, hence {n ∈ N : n /∈ U} ∈ I. This means
that the sequence {n}n∈N in Y satisfies n

I−→ 0. But {n ∈ N : n /∈ f(X1)} =
{2k + 1, k ∈ ω} /∈ I. Thus f(X1) is not an I-sequential neighborhood of 0 in
Y . □

Problem 3.13. For some maximal ideal J of N and an J -quotient mapping
f, does it satisfy the condition (6) of Theorem 3.10?

4. On Spaces with I-cs′-Networks

In this section, we introduce the concepts of I-cs-networks, I-cs′-networks
and I-wcs′-networks for a space X; and obtain that if X is an J -FU space
with a point-countable J -cs′-network, then X is a meta-Lindelöf space, for a
maximal ideal J of N.

Definition 4.1 ([13]). Let I be an ideal on N, X be a topological space and
P be a cover of X.

(1) P is a network of X if whenever x ∈ U with U open in X, then
x ∈ P ⊆ U for some P ∈ P.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 242



On I-quotient mappings and I-cs′-networks under a maximal ideal

(2) P is called an I-cs-network of X if whenever {xn}n∈N is a sequence in
X I-converging to a point x ∈ U with U open in X, then {xn}n∈N is
I-eventually in P and x ∈ P ⊆ U for some P ∈ P.

(3) P is called an I-cs′-network of X if whenever {xn}n∈N is a sequence
in X I-converging to a point x ∈ U with U open in X, then there is
P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U.

(4) P is called an I-wcs′-network of X if whenever {xn}n∈N is a sequence
in X I-converging to a point x ∈ U with U open in X, then there is
P ∈ P and some n0 ∈ N such that {xn0} ⊆ P ⊆ U.

Obviously, I-cs-networks ⇒ I-cs′-networks ⇒ I-wcs′-networks ⇒ networks.
Definition 4.2. Let J be a maximal ideal of N and X be a topological space.
U is said to be J -sn-cover of X, if {(U)J -s : U ∈ U} is a cover of X.
Theorem 4.3. Each I-cs-network is preserved by an I-covering mapping.
Proof. Let f : X → Y be an I-covering mapping and P be an I-cs-network of
X. Suppose that {yn}n∈N is a sequence I-converging to a point y ∈ U with U
open in Y . Since f is an I-covering mapping, there exist a sequence {xn}n∈N of
points xn ∈ f−1(yn) for all n ∈ N and x ∈ f−1(y) such that xn

I−→ x. Since P is
an I-cs-network of X, there is some P ∈ P such that {xn}n∈N is I-eventually in
P and x ∈ P ⊆ f−1(U). Thus there is E ∈ I such that {n ∈ N : xn /∈ P} ⊆ E.
Note that {n ∈ N : yn /∈ f(P)} ⊆ {n ∈ N : xn /∈ P} ⊆ E, hence yn ∈ f(P) for
all n ∈ N \ E, i.e. {yn}n∈N is I-eventually in f(P) and y ∈ f(P) ⊆ U. This
means that f(P) = {f(P) : P ∈ P} is an I-cs-network of Y . □
Corollary 4.4. Each I-cs′-network is preserved by an I-covering mapping.
Theorem 4.5. Each I-wcs′-network is preserved by an I-quotient mapping.
Proof. Let f : X → Y be an I-quotient mapping and P be an I-wcs′-network
of X. Suppose that {yn}n∈N is a sequence I-converging to a point y ∈ U with U
open in Y . Since f is an I-quotient mapping, there is a sequence {xi}i∈N in X
such that {xi : i ∈ N} ⊆ f−1({yn : n ∈ N}) and xi

I−→ x /∈ f−1({yn : n ∈ N}).
And because P is an I-wcs′-network of X, there is some P0 ∈ P and i0 ∈ N
such that {xi0} ⊆ P0 ⊆ f−1(U). And hence {f(xi0)} = {yn0} ⊆ f(P0) ⊆ U for
some n0 ∈ N. This implies that f(P) = {f(P) : P ∈ P} is an I-wcs′-network
of Y . □
Lemma 4.6. Let J be a maximal ideal of N and P be a family of subsets of
X. Then P is an J -cs′-network of X if and only if, whenever U is an open
neighborhood of x,

"
{P ∈ P : x ∈ P ⊆ U} is an J -sequential neighborhood of

x.

Proof. Necessity: Let U be an open neighborhood of x. If
"
{P ∈ P : x ∈

P ⊆ U} is not an J -sequential neighborhood of x, then there is a sequence
{xn}n∈N such that xn

J−→ x and xn /∈
"
{P ∈ P : x ∈ P ⊆ U} for each n ∈ N.

Since P is an J -cs′-network of X, there is P0 ∈ P and n0 ∈ N such that
{x, xn0} ⊆ P0 ⊆ U, a contradiction.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 243



X. Zhou

Sufficiency: Suppose that xn
J−→ x ∈ U ∈ τX and

"
{P ∈ P : x ∈ P ⊆ U} is

an J -sequential neighborhood of x. Then {xn}n∈N is J -eventually in
"
{P ∈

P : x ∈ P ⊆ U}. Hence there exists n0 ∈ N such that xn0 ∈
"
{P ∈ P : x ∈

P ⊆ U}. And hence there is P0 ∈ P such that xn0 ∈ P0 and x ∈ P0 ⊆ U. Thus
{x, xn0} ⊆ P0 ⊆ U. This means that P is an J -cs′-network of X. □

Theorem 4.7. Let J be a maximal ideal of N and a space X be of a point-
countable J -cs′-network. Then each open cover of X has a point-countable
J -sn refinement.

Proof. Suppose that P is a point-countable J -cs′-network for a space X. Let
U = {Uα}α<γ be an open cover of X, where γ is an ordinal. For each α < γ,
put

Vα =
#

{P ∈ P : P ⊆ Uα, P ∕⊆ Uβ if β < α}.

Clearly, Vα ⊆ Uα. Next we shall show that the family V = {Vα}α<γ is a point-
countable J -sn-cover of X. For each x ∈ X, let α(x) = min{α < γ : x ∈ Uα}.
Then x ∈ Uα(x) and
#

{P ∈ P : x ∈ P ⊆ Uα(x)} ⊆
#

{P ∈ P : P ⊆ Uα(x), P ∕⊆ Uβ if β < α(x)}.

Since P is an J -cs′-network for a space X, it follows from Lemma 4.6 that

x ∈ (
#

{P ∈ P : x ∈ P ⊆ Uα(x)})J -s

⊆ (
#

{P ∈ P : P ⊆ Uα(x), P ∕⊆ Uβ if β < α(x)})J -s
= (Vα(x))J -s.

This means that V = {Vα}α<γ is an J -sn-cover of X.
We claim that V is point-countable. Suppose, to the contrary, that there

exist a point x ∈ X and an uncountable subset Γ of γ such that x ∈ Vα for
each α ∈ Γ. Hence there is Pα ∈ P such that x ∈ Pα ⊆ Uα and Pα ∕⊆ Uβ
for β < α. Since P is a point-countable family and Γ is an uncountable set,
there are α, β ∈ Γ, α ∕= β such that Pα = Pβ. Assume that β < α, then
Uβ ⊇ Pβ = Pα ∕⊆ Uβ, a contradiction. □

Definition 4.8.

(1) A space X is called I-Fréchet-Urysohn (or shortly, I-FU) space, if for
each A ⊂ X and each x ∈ A, there exists a sequence in A I-converging
to the point x in X [20].

(2) A space X is called a meta-Lindelöf space if each open cover of X has
a point-countable open refinement [13].

Corollary 4.9. Let J be a maximal ideal of N. If X is an J -FU space with
a point-countable J -cs′-network, then X is a meta-Lindelöf space.

Proof. X is an J -FU space ⇔ A = [A]J -s for each A ⊆ X ⇔ intA = (A)J -s
for each A ⊆ X. □

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 244



On I-quotient mappings and I-cs′-networks under a maximal ideal

Theorem 4.10. Let J be a maximal ideal of N. The following are equivalent
for a space X.

(1) J -sX is an J -Fréchet-Urysohn space.
(2) clJ -sX(A) = [A]J -s, for each A ⊆ X.
(3) [A]J -s is J -closed in X, for each A ⊆ X.
(4) (A)J -s is J -open in X, for each A ⊆ X.

Proof. Since the spaces X and J -sX have the same J -convergent sequences, by
the Definition 4.8 and Proposition 3.8, it follows that (1) ⇔ (2) and (3) ⇔ (4).
Hence, it suffices to show that (2) ⇔ (3). If clJ -sX(A) = [A]J -s, then [A]J -s
is closed in J -sX, and hence [A]J -s is J -closed in X, for each A ⊆ X. On
the other hand, if [A]J -s is J -closed in X, then [A]J -s is closed in J -sX, and
further clJ -sX(A) = [A]J -s, for each A ⊆ X. □

Acknowledgements. My gratitude goes to Professor Shou Lin, for his friendly
encouragement and inspiring suggestions.

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