@ Appl. Gen. Topol. 22, no. 2 (2021), 417-434doi:10.4995/agt.2021.15231
© AGT, UPV, 2021

Small and large inductive dimension for ideal

topological spaces

F. Sereti

University of Patras, Department of Mathematics, 26504, Patra, Greece (seretifot@gmail.com)

Communicated by P. Das

Abstract

Undoubtedly, the small inductive dimension, ind, and the large in-
ductive dimension, Ind, for topological spaces have been studied exten-
sively, developing an important field in Topology. Many of their proper-
ties have been studied in details (see for example [1,4,5,9,10,18]). How-
ever, researches for dimensions in the field of ideal topological spaces
are in an initial stage. The covering dimension, dim, is an exception
of this fact, since it is a meaning of dimension, which has been stud-
ied for such spaces in [17]. In this paper, based on the notions of the
small and large inductive dimension, new types of dimensions for ideal
topological spaces are studied. They are called ∗-small and ∗-large
inductive dimension, ideal small and ideal large inductive dimension.
Basic properties of these dimensions are studied and relations between
these dimensions are investigated.

2010 MSC: 54F45; 54A05; 54A10.

Keywords: small inductive dimension; large inductive dimension; ideal
topological space.

1. Introduction and preliminaries

The Dimension Theory is a developing branch of Topology, which attracts
the interest of many researches (see for example [1, 2, 4–13, 18]). Especially,
the covering dimension, dim, the small inductive dimension, ind, and the large
inductive dimension, Ind, are three main topological dimensions, which have

Received 09 March 2021 – Accepted 06 June 2021

http://dx.doi.org/10.4995/agt.2021.15231


F. Sereti

been studied extensively, and many results in various classes of topological
spaces have been proved.

Simultaneously, the notion of ideal leads to an important chapter in Topology
(see for example [3,14,15,19]). The main notion of ideal topological space was
studied in Kuratowski’s monograph [16]. However, the notion of topological
dimension has not been investigated under the prism of ideals. The covering
dimension is an exception. In the paper [17], the meaning of the ideal covering
dimension is inserted and studied in details. Thus, in this paper new notions of
inductive dimensions for ideal topological spaces are introduced and studied.
They are called ∗-small and ∗-large inductive dimension, ideal small and ideal
large inductive dimension.

Especially, in Section 2, we insert the meanings of the so-called ∗-small in-
ductive dimension, ind

∗
, and ∗-large inductive dimension, Ind

∗
, for an arbitrary

ideal topological space and study basic results. In Sections 3, we insert and
study the meanings of the ideal small inductive dimension, I - ind, and the ideal
large inductive dimension, I - Ind, and finally, in Section 4, we study additional
properties of the ideal topological dimensions.

It is considered to be necessary, to recall the main notions and notations
that will be used in the rest of this study. Especially, the notion of the ideal
topological space and the known meanings of the small inductive dimension
and the large inductive dimension are presented. The standard notation of
Dimension Theory is referred to [1,5,18].

A nonempty family I of subsets of a set X is called an ideal on X if it
satisfies the following properties:

(1) If A ∈ I and B ⊆ A, then B ∈ I.
(2) If A, B ∈ I, then A ∪ B ∈ I.

A topological space (X, τ) with an ideal I is called an ideal topological space
and is denoted by (X, τ, I). In [15] the authors defined a new topology τ∗ on
X in terms of the Kuratowski closure operator cl∗. It is known that the family

β∗ = {U \ I : U ∈ τ, I ∈ I}

is a basis for τ∗ and the topology τ∗ is finer than τ. Especially, if I = {∅},
then τ∗ = τ and if I = P(X), then τ∗ is the discrete topology.

In what follows, by an open set (resp. closed set), we mean an open set (resp.
closed set) in the topology τ. If a set U is open in the topology τ∗, then we
say that U is a ∗-open set. Similarly, we define ∗-closed sets. If A ⊆ X, then
BdX(A) and Bd

∗
X(A) will denote the boundary of A in (X, τ) and (X, τ

∗),
respectively. Similarly, ClX(A) and Cl

∗
X(A) will denote the closure of A in

(X, τ) and (X, τ∗), respectively. Also, If denotes the ideal of all finite subsets
of X.

Definition 1.1. The small inductive dimension of a topological space X, de-
noted by ind(X), is defined as follows:

(i) ind(X) = −1, if X = ∅.

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Small and large inductive dimension for ideal topological spaces

(ii) ind(X) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ X and for
every open subset V of X with x ∈ V , there exists an open subset U
of X such that

x ∈ U ⊆ V and ind(BdX(U)) 6 k − 1.

(iii) ind(X) = k, where k ∈ {0, 1, . . .}, if ind(X) 6 k and ind(X) 
 k − 1.
(iv) ind(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which

ind(X) 6 k is true.

Definition 1.2. The large inductive dimension of a topological space X, de-
noted by Ind(X), is defined as follows:

(i) Ind(X) = −1, if X = ∅.
(ii) Ind(X) 6 k, where k ∈ {0, 1, . . .}, if for every pair (F, V ) of subsets of

X, where F is closed, V is open and F ⊆ V , there exists an open set
U of X such that

F ⊆ U ⊆ V and Ind(BdX(U)) 6 k − 1.

(iii) Ind(X) = k, where k ∈ {0, 1, . . .}, if Ind(X) 6 k and Ind(X) 
 k − 1.
(iv) Ind(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which

Ind(X) 6 k is true.

2. The small inductive dimension and the large inductive
dimension for ideal topological spaces

In this section, based on the notion of the topology τ∗, the ∗-small and ∗-
large inductive dimension are defined for an ideal topological space (X, τ, I)
and basic properties are studied.

Definition 2.1. The ∗-small inductive dimension of an ideal topological space
(X, τ, I), denoted by ind∗(X), is defined as follows:

(i) ind∗(X) = −1, if X = ∅.
(ii) ind∗(X) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ X and for

every ∗-open subset V of X with x ∈ V , there exists a ∗-open subset
U of X such that

x ∈ U ⊆ V and ind∗(Bd∗X(U)) 6 k − 1.

(iii) ind∗(X) = k, where k ∈ {0, 1, . . .}, if ind∗(X) 6 k and ind∗(X) 
 k−1.
(iv) ind

∗
(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which

ind∗(X) 6 k is true.

Definition 2.2. The ∗-large inductive dimension of an ideal topological space
(X, τ, I), denoted by Ind∗(X), is defined as follows:

(i) Ind∗(X) = −1, if X = ∅.
(ii) Ind∗(X) 6 k, where k ∈ {0, 1, . . .}, if for every pair (F, V ) of subsets of

X, where F is ∗-closed, V is ∗-open and F ⊆ V , there exists a ∗-open
set U of X such that

F ⊆ U ⊆ V and Ind∗(Bd∗X(U)) 6 k − 1.

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F. Sereti

(iii) Ind∗(X) = k, where k ∈ {0, 1, . . .}, if Ind∗(X) 6 k and Ind∗(X) 
 k−1.
(iv) Ind∗(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for which

Ind∗(X) 6 k is true.

We observe that the dimensions ind∗(X) and Ind∗(X) are the small and the
large inductive dimension, respectively, of the topological space (X, τ∗). The
following result will be useful in the rest of this study.

Theorem 2.3.

(1) For any subset A of X, ind∗(A) 6 ind∗(X).
(2) For any ∗-closed subset A of X, Ind∗(A) 6 Ind∗(X).

Proof. Based on Definition 2.1 and Definition 2.2, (1) and (2) can be proved
by induction on the dimension ind∗(X) and Ind∗(X), respectively. �

However, the following examples prove that these dimensions are different
to each other and different from the small inductive dimension, ind, and the
large inductive dimension, Ind.

Example 2.4. Let X = {a, b, c} with the topology τ = {∅, {b, c}, X}. If we
consider the ideal I = {∅, {a}}, then ind∗(X) = 1 and Ind∗(X) = 0.

Example 2.5.

(1) We consider the set X = {a, b, c, d} with the topology

τ = {∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, X}.

Then ind(X) = 1. However, if we consider the ideal I = {∅, {a}, {b}, {a, b}},
then the space (X, τ∗) is the discrete space and thus, ind∗(X) = 0.

(2) We consider the set X = {a, b, c} with the topology τ = {∅, X}. Then
ind(X) = 0. However, if we consider the ideal I = {∅, {b}}, then
ind∗(X) = 1.

Example 2.6.

(1) We consider the set X = {a, b, c, d} with the topology

τ = {∅, {a}, {a, b}, {a, d}, {a, b, d}, {a, c, d}, X}.

Then Ind(X) = 1. However, if we consider the ideal I = {∅, {a}, {d}, {a, d}},
then the space (X, τ∗) is the discrete space and thus, Ind

∗
(X) = 0.

(2) We consider the set X = {a, b, c, d} with the topology

τ = {∅, {a}, {b}, {a, b}, {a, b, d}, X}.

Then Ind(X) = 0. However, if we consider the ideal I = {∅, {b}, {d}, {b, d}},
then Ind∗(X) = 1.

Remark 2.7. For any ideal topological space (X, τ, I) for which τ = τ∗ we have

(1) ind(X) = ind∗(X) and
(2) Ind(X) = Ind

∗
(X).

However, the converse of Remark 2.7 does not always hold and the following
examples prove this claim.

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Small and large inductive dimension for ideal topological spaces

Example 2.8.

(1) Let X = {a, b, c} with the topology τ = {∅, {b, c}, X}. Then ind(X) =
1. If we consider the ideal I = {∅, {b}}, then ind∗(X) = 1 but τ 6= τ∗.

(2) Let X = {a, b, c, d} with the topology

τ = {∅, {a}, {b}, {a, b}, {a, b, d}, X}.

Then Ind(X) = 0. However, if we consider the ideal I = {∅, {b}}, then
Ind

∗
(X) = 0 but τ 6= τ∗.

Moreover, in the following propositions we can prove further relations be-
tween these dimensions.

Proposition 2.9. For every ideal topological space (X, τ, I), where I ⊆ τc,
the following are satisfied:

(1) ind(X) = ind∗(X) and
(2) Ind(X) = Ind

∗
(X).

Proof. Since I ⊆ τc, we have that β∗ ⊆ τ. Therefore, τ = τ∗ and by Remark
2.7 we have that

ind(X) = ind∗(X) and Ind(X) = Ind∗(X).

�

Proposition 2.10. For every ideal topological T1-space (X, τ, I), where I ⊆
If , the following are satisfied:

(1) ind(X) = ind∗(X) and
(2) Ind(X) = Ind∗(X).

Proof. Since the ideal topological space (X, τ, I) is T1, every I ∈ I is closed in
(X, τ). Therefore, τ = τ∗ and by Proposition 2.9 we have that

ind(X) = ind∗(X) and Ind(X) = Ind∗(X). ✷

Let (X, τ, I) be an ideal topological space. If β∗ is a topology on X (and
hence τ∗ = β∗), then the ideal I is called τ-simple [14]. Also, the ideal I is
called τ-codense if I ∩ τ = {∅}, that is each member of I has empty interior
with respect to the topology τ [3]. Moreover, we state that if A ⊆ X, then the
family

IA = {A ∩ I : I ∈ I}

is an ideal on A. So, we can consider the ideal topological space (A, τA, IA),
where τA is the subspace topology on A. The topology (τA)

∗ is equal to the
subspace topology (τ∗)A on A [15]. �

Proposition 2.11. Let (X, τ, I) be an ideal topological space. If I is τ-simple
and τ-codense and for each closed subset F of X the ideal IF is τF -codense,
then

ind(X) 6 ind∗(X).

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F. Sereti

Proof. Clearly, if ind∗(X) = −1 or ind∗(X) = ∞, then the inequality holds.
We suppose that the inequality holds for all integers m < k and we shall prove
it for k. Let ind∗(X) = k, x ∈ X and V be an open set in X with x ∈ V .
Then V is a ∗-open set in X. Since ind∗(X) = k, there exists a ∗-open set U
in X with x ∈ U ⊆ V and ind

∗
(Bd

∗
X(U)) 6 k − 1. Also, since I is τ-simple,

U = W \ I, where W ∈ τ and I ∈ I.
We consider the open set V ∩ W in X. Then x ∈ U ⊆ V ∩ W ⊆ V . It

suffices to prove that ind(BdX(V ∩ W)) 6 k − 1. Let Y = BdX(V ∩ W) with
the subspace topology τY and the ideal IY . We have that IY is τY -simple, as
β∗Y is a topology on Y , and τY -codense, as Y is a closed subset of X. Also, we
should prove that for every closed set K in Y , the ideal IK is τK-codense. Let
K be a closed set in Y . Since Y is closed in X, K is also closed in X and thus,
IK is τK-codense. Moreover, we observe that

BdX(V ∩ W) ⊆ Bd
∗
X(U).

Indeed, we suppose that there exists y ∈ BdX(V ∩W) such that y /∈ Bd
∗
X(U).

We have that y ∈ X \ (V ∩ W) and thus, y ∈ X \ U. Therefore, y /∈ Cl∗X(U).
That is, there exists a ∗-open set O in X with y ∈ O and O ∩ U = ∅. Since
I is a τ-simple ideal, we have that O = P \ J, where P ∈ τ and J ∈ I.
Thus, P is an open set in X with y ∈ P . Since y ∈ ClX(V ∩ W), we have that
P ∩(V ∩W) 6= ∅ or equivalently, (P ∩W)∩V 6= ∅. Since (P ∩W)∩V ⊆ P ∩W ,
we have that P ∩ W 6= ∅. Also, the relation O ∩ U = ∅ implies the relation
(P \ J) ∩ (W \ I) = ∅ and thus, P ∩ W ⊆ I ∪ J. Since I is an ideal and
I, J ∈ I, we have that I ∪ J ∈ I. Thus, the member I ∪ J of the ideal I has
non empty interior with respect to the topology τ, which is a contradiction as
I is τ-codense. Thus, BdX(V ∩ W) ⊆ Bd

∗
X(U).

By the Subspace Theorem for the dimension ind∗ (see Theorem 2.3), we
have that

ind
∗
(BdX(V ∩ W)) 6 ind

∗
(Bd

∗
X(U)) 6 k − 1

and by inductive hypothesis, we have that

ind(BdX(V ∩ W)) 6 ind
∗(BdX(V ∩ W)).

Therefore, ind(BdX(V ∩ W)) 6 k − 1. Thus, ind(X) 6 k. �

Proposition 2.12. Let (X, τ, I) be an ideal topological space. If I is τ-simple
and τ-codense and for each closed subset F of X the ideal IF is τF -codense,
then

Ind(X) 6 Ind∗(X).

Proof. It is similar to the proof of Proposition 2.11. �

Proposition 2.13. Let (X, τ, I) be an ideal topological space and U be a ∗-open
set in X. Then

IBd∗
X
(U) ⊆ (τ

∗
Bd∗

X
(U))

c.

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Small and large inductive dimension for ideal topological spaces

Proof. Let A ∈ IBd∗
X
(U). Then there exists I ∈ I such that A = Bd

∗
X(U) ∩ I.

Since I is closed in the space (X, τ∗) [15], that is ∗-closed in X, A is ∗-closed
in Bd∗X(U), proving the relation of the proposition. �

3. The ideal small inductive dimension I - ind and the ideal large
inductive dimension I - Ind for ideal topological spaces

In this section, the notions of the ideal small inductive dimension, I - ind,
and the ideal large inductive dimension, I - Ind, of an ideal topological space
(X, τ, I), are defined, combining the topologies τ and τ∗, and basic properties
of these dimensions are investigated.

Definition 3.1. The ideal small inductive dimension of an ideal topological
space (X, τ, I), denoted by I - ind(X), is defined as follows:

(i) I - ind(X) = −1, if X = ∅.
(ii) I - ind(X) 6 k, where k ∈ {0, 1, . . .}, if for every element x ∈ X and

for every open subset V of X with x ∈ V , there exists a ∗-open subset
U of X with

x ∈ U ⊆ V and IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

(iii) I - ind(X) = k, where k ∈ {0, 1, . . .}, if I - ind(X) 6 k and I - ind(X) 

k − 1.

(iv) I - ind(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for
which I - ind(X) 6 k is true.

It is observed that the ideal small inductive dimension I - ind is different from
the dimensions ind and ind∗ and the following examples prove this assertion.

Example 3.2.

(1) Let k > 1. We consider the set Xk = {0, 1, 2, . . ., k} and the topo-
logy generated by the family {∅, {0}, {0, 1}, . . ., {0, 1, . . . , k}}. Then
ind(Xk) = k. If we consider the ideal I, consisting of all subsets of X,
then τ∗ is the discrete topology and therefore, I - ind(X) = 0.

(2) We consider the indiscrete space (R, τ) and the ideal I = {I ⊆ R : 0 /∈
I}. Then τ∗ = {∅}∪{U ⊆ R : 0 ∈ U}, ind∗(R) = 1 and I - ind(R) = 0.

Lemma 3.3. Let X be a non-empty set, A, B subsets of X such that A ⊆ B
and I an ideal on X. Then (IB)A = IA.

Theorem 3.4. If (X, τ, I) is an ideal topological space and A is a subset of
X, then

IA-ind(A) 6 I - ind(X).

Proof. Obviously, if I - ind(X) = −1 or I - ind(X) = ∞, then the inequality
holds. We suppose that the inequality holds for all integers m < k and we shall
prove it for k. Let I - ind(X) = k, x ∈ A and VA be an open set in A with
x ∈ VA. Then there exists an open set V in X such that VA = V ∩ A. Clearly,
x ∈ V . Since I - ind(X) = k, there exists a ∗-open set U in X with x ∈ U ⊆ V
and IBd∗

X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

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F. Sereti

We consider the ∗-open set UA = U ∩ A in A. Then x ∈ UA ⊆ VA. We shall
prove that

(IA)Bd∗
A
(UA)-ind(Bd

∗
A(UA)) 6 k − 1.

By Lemma 3.3, it suffices to prove that

IBd∗
A
(UA)-ind(Bd

∗
A(UA)) 6 k − 1.

Since

Bd∗A(UA) = Bd
∗
A(U ∩ A) ⊆ A ∩ Bd

∗
X(U) ⊆ Bd

∗
X(U),

Bd∗A(UA) is a subset of Bd
∗
X(U) and by inductive hypothesis, we have that

(IBd∗
X
(U))Bd∗

A
(UA)-ind(Bd

∗
A(UA)) 6 k − 1.

Moreover, applying Lemma 3.3 we have that

(IBd∗
X
(U))Bd∗

A
(UA) = IBd∗A(UA).

Thus, IBd∗
A
(UA)-ind(Bd

∗
A(UA)) 6 k − 1 and so IA-ind(A) 6 k. �

Proposition 3.5. For any ideal topological space (X, τ, I) we have that

I - ind(X) 6 min{ind(X), ind∗(X)}.

Proof. Firstly, we shall prove that I - ind(X) 6 ind(X). Clearly, if ind(X) =
−1 or ind(X) = ∞, then the inequality of the proposition holds. We suppose
that the inequality holds for all integers m < k and we shall prove it for k. Let
ind(X) = k, x ∈ X and V be an open set in X with x ∈ V . Since ind(X) = k,
there exists an open set U in X with x ∈ U ⊆ V and ind(BdX(U)) 6 k − 1.
Then U is a ∗-open set and by inductive hypothesis, we have that

IBdX (U)-ind(BdX(U)) 6 k − 1.

Since Bd∗X(U) ⊆ BdX(U), by Theorem 3.4 we have

(IBdX (U))Bd∗X(U)-ind(Bd
∗
X(U)) 6 k − 1.

By Lemma 3.3 we have that

IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

and therefore, I - ind(X) 6 k.

We shall prove that I - ind(X) 6 ind∗(X). Clearly, if ind∗(X) = −1 or
ind∗(X) = ∞, then the inequality of the proposition holds. We suppose that
the inequality holds for all integers m < k and we shall prove it for k. Let
ind∗(X) = k, x ∈ X and V be an open set in X with x ∈ V . Then V is also
a ∗-open set in X. Since ind∗(X) = k, there exists a ∗-open set U in X with
x ∈ U ⊆ V and ind∗(Bd∗X(U)) 6 k − 1. Then by inductive hypothesis we have
that

IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1

and thus, I - ind(X) 6 k. �

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Small and large inductive dimension for ideal topological spaces

Proposition 3.6. Let (X, τ, I) be an ideal topological space and k ∈ {0, 1, . . .}.
If there exists a base B of (X, τ∗) such that IBd∗

X
(U)-ind(Bd

∗
X(U)) 6 k − 1, for

every U ∈ B, then I - ind(X) 6 k.

In what follows, we show characterizations for the ideal small inductive di-
mension, assuming that (X, τ) is a regular space. We state firstly that a subset
L of the space (X, τ∗) is called a ∗-partition between two subsets A and B
of X if there exist ∗-open sets U, V with A ⊆ U, B ⊆ V , U ∩ V = ∅ and
X \ L = U ∪ V .

Proposition 3.7. Let (X, τ, I) be an ideal topological space, where (X, τ) is
a regular space. Then I - ind(X) 6 k, where k ∈ {0, 1, . . .}, if and only if for
every point x ∈ X and for every closed subset F of X with x /∈ F, there exists
a ∗-partition L between {x} and F with IL-ind(L) 6 k − 1.

Proof. Let k ∈ {0, 1, . . .}. Firstly, we suppose that I - ind(X) 6 k. Let x ∈ X
and F be a closed subset of X with x /∈ F . Then we consider the set V = X\F .
Then V is an open set in X with x ∈ V . Since (X, τ) is a regular space, there
exists an open set V1 (and thus a ∗-open set) in X such that

x ∈ V1 ⊆ ClX(V1) ⊆ V.

Since I - ind(X) 6 k and x ∈ V1, there exists a ∗-open set U in X with
x ∈ U ⊆ V1 and IBd∗

X
(U)-ind(Bd

∗
X(U)) 6 k − 1. Then

x ∈ Cl∗X(U) ⊆ Cl
∗
X(V1) ⊆ ClX(V1) ⊆ V

and the set L = Bd∗X(U) is a ∗-partition between the sets {x} and F such that
IL-ind(L) 6 k − 1.

Conversely, we shall prove that I - ind(X) 6 k. Let x ∈ X and Vx be an
open set in X with x ∈ Vx. We set F = X \ Vx. Then F is a closed subset of
X with x /∈ F . By assumption there exists a ∗-partition L between {x} and F
with IL-ind(L) 6 k − 1. That is, there exist ∗-open sets U and V in X with
x ∈ U, F ⊆ V , U ∩V = ∅ and X \L = U ∪V . Since Bd∗X(U) ⊆ L, by Theorem
3.4, we have that

(IL)Bd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1

and by Lemma 3.3,

IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

Therefore, I - ind(X) 6 k. �

Corollary 3.8. A non-empty ideal topological space X is ideal zero dimensional
(with respect to the ideal small inductive dimension), where (X, τ) is a regular
space, if and only if for every point x ∈ X and each closed subset F of X such
that x /∈ F the empty set is a ∗-partition between {x} and F.

Proof. It follows directly by Proposition 3.7. �

Proposition 3.9. If (X, τ, I) is an ideal topological space, where (X, τ) is a
regular space, and k ∈ {0, 1, . . .}, then the following statements are equivalent:

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F. Sereti

(1) I - ind(X) 6 k,
(2) for every open neighborhood V of a point x in X, there exists a ∗-open

set U and a ∗-closed set E in X such that

x ∈ U ⊆ E ⊆ V and IE\U -ind(E \ U) 6 k − 1,
(3) for every open neighborhood V of a point x in X, there exists a ∗-open

set U in X such that

x ∈ U ⊆ Cl∗X(U) ⊆ V and IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

Proof. We prove the implication (1) ⇒ (2). Let I - ind(X) 6 k and V be an
open neighborhood of a point x in X. Then by Proposition 3.7, there exists
a ∗-partition L between {x} and X \ V with IL-ind(L) 6 k − 1. We consider
disjoint ∗-open sets U and G in X with x ∈ U, X \ V ⊆ G and X \ L = U ∪ G.
We set E = X \ G. Then E is a ∗-closed set in X with x ∈ U ⊆ E ⊆ V and
E \ U = L, completing the proof of this assertion.

Next, we prove the implication (2) ⇒ (3). Let V be an open neighborhood
of a point x in X. By (2), there exists a ∗-open set U and a ∗-closed set E
in X with x ∈ U ⊆ E ⊆ V and IE\U -ind(E \ U) 6 k − 1. We observe that
Cl∗X(U) ⊆ E and Bd

∗
X(U) ⊆ E \ U. Thus, by Theorem 3.4, we have that

(IE\U )Bd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1

and by Lemma 3.3 we have that

IBd∗
X
(U)-ind(Bd

∗
X(U)) 6 k − 1.

Finally, we prove the implication (3) ⇒ (1). Let F be a closed set in X and
x ∈ X with x /∈ F . We put V = X \ F . Then V is an open neighborhood of
x. By (3), there exists a ∗-open set U in X with x ∈ U ⊆ Cl∗X(U) ⊆ V and
IBd∗

X
(U)-ind(Bd

∗
X(U)) 6 k − 1. Then U and X \ Cl

∗
X(U) are disjoint ∗-open

sets in X, containing x and F , respectively, and

X \ (U ∪ (X \ Cl∗X(U))) = (X \ U) ∩ Cl
∗
X(U) = Bd

∗
X(U).

Thus, Bd∗X(U) is a ∗-partition between {x} and F and therefore, by Proposition
3.7, we have that I - ind(X) 6 k. �

In the next results we study the ideal small inductive dimension of the
topological sum.

Lemma 3.10. Let {Xs : s ∈ S} be a set of pairwise disjoint topological spaces
and

⊕
s∈S Xs be their topological sum. Let also for each s ∈ S, Is be an ideal

of Xs. Then

I = {
⋃

Es : Es ∈ Is for each s ∈ S}

is an ideal of
⊕

s∈S Xs.

Theorem 3.11. Let {Xs : s ∈ S} be a set of pairwise disjoint topological
spaces and

⊕
s∈S Xs be their topological sum. Let also for each s ∈ S, Is be

an ideal of Xs. Then I - ind(
⊕

s∈S Xs) 6 k, where k ∈ {0, 1, . . .} and I is the
ideal of Lemma 3.10, if and only if Is-ind(Xs) 6 k for each s ∈ S.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 426



Small and large inductive dimension for ideal topological spaces

Proof. Let k ∈ {0, 1, . . .}. We suppose that I - ind(
⊕

s∈S Xs) 6 k and we shall
prove that Is-ind(Xs) 6 k for each s ∈ S. Let s ∈ S. In the proof we shall
write Y =

⊕
s∈S Xs. Let x ∈ Xs and V be an open subset of Xs with x ∈ V .

Then V is an open subset of Y . Since I - ind(Y ) 6 k, there exists a ∗-open
subset U of Y such that x ∈ U ⊆ V and

IBd∗
Y
(U)-ind(Bd

∗
Y (U)) 6 k − 1.

We observe that IXs = Is and the set U ∩ Xs is a ∗-open subset of Xs with
x ∈ U ∩ Xs ⊆ U ⊆ V . It suffices to prove that

(IXs)Bd∗
Xs

(U∩Xs)-ind(Bd
∗
Xs

(U ∩ Xs)) 6 k − 1

or by Lemma 3.3,

IBd∗
Xs

(U∩Xs)-ind(Bd
∗
Xs

(U ∩ Xs)) 6 k − 1.

We have that Bd∗Xs(U ∩Xs) ⊆ Bd
∗
Y (U). Therefore, by Theorem 3.4 we have

that

(IBd∗
Y
(U))Bd∗

Xs
(U∩Xs)-ind(Bd

∗
Xs

(U ∩ Xs)) 6 IBd∗
Y
(U)-ind(Bd

∗
Y (U))

and by Lemma 3.3 we have that

IBd∗
Xs

(U∩Xs)-ind(Bd
∗
Xs

(U ∩ Xs)) 6 IBd∗
Y
(U)-ind(Bd

∗
Y (U))

and therefore,

IBd∗
Xs

(U∩Xs)-ind(Bd
∗
Xs

(U ∩ Xs)) 6 k − 1.

Conversely, we suppose that Is-ind(Xs) 6 k for each s ∈ S and we shall
prove that I - Ind(Y ) 6 k. Let x ∈ Y and V be an open subset of Y with
x ∈ V . Then there exists s ∈ S such that x ∈ Xs and the set V ∩Xs is an open
subset of Xs with x ∈ V ∩ Xs. Since Is-ind(Xs) 6 k, there exists a ∗-open
subset Us of Xs such that x ∈ Us ⊆ V ∩ Xs ⊆ V and

(Is)Bd∗
Xs

(Us)-ind(Bd
∗
Xs

(Us)) 6 k − 1

or equivalently,

(IXs)Bd∗
Xs

(Us)-ind(Bd
∗
Xs

(Us)) 6 k − 1

and by Lemma 3.3,

IBd∗
Xs

(Us)-ind(Bd
∗
Xs

(Us)) 6 k − 1.

The set Us is ∗-open subset of Y and it suffices to prove that

IBd∗
Y
(Us)-ind(Bd

∗
Y (Us)) 6 k − 1.

We have that Bd∗Xs(Us) = Bd
∗
Y (Us) and thus,

IBd∗
Y
(Us)-ind(Bd

∗
Y (Us)) 6 k − 1.

�

In what follows, we shall define and study the ideal large inductive dimension
I - Ind of an ideal topological space (X, τ, I).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 427



F. Sereti

Definition 3.12. The ideal large inductive dimension of an ideal topological
space (X, τ, I), denoted by I - Ind(X), is defined as follows:

(i) I - Ind(X) = −1, if X = ∅.
(ii) I - Ind(X) 6 k, where k ∈ {0, 1, . . .}, if for every pair (F, V ) of subsets

of X, where F is closed, V is open and F ⊆ V , there exists a ∗-open
subset U of X such that

F ⊆ U ⊆ V and IBd∗
X
(U)-Ind(Bd

∗
X(U)) 6 k − 1.

(iii) I - Ind(X) = k, where k ∈ {0, 1, . . .}, if I - Ind(X) 6 k and I - Ind(X) 

k − 1.

(iv) I - Ind(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for
which I - Ind(X) 6 k is true.

We observe that the ideal large inductive dimension is different from the
dimensions Ind and Ind∗ and the following examples prove this assertion.

Example 3.13.

(1) We consider the space X = {a, b, c, d} with the topology

τ = {∅, {a}, {a, b}, {a, b, c}, {a, b, d}, X}

and the ideal I = {∅, {a}, {b}, {a, b}}. Then Ind(X) = 1 and I - Ind(X) =
0.

(2) We consider the space X = {a, b, c, d, e} with the topology

τ = {∅, {a}, {a, b}, {a, c}, {a, b, c}, {a, b, c, e}, X}

and the ideal I = {∅, {c}, {e}, {c, e}}. Then Ind∗(X) = 2 and I - Ind(X) =
0.

Theorem 3.14. If (X, τ, I) is an ideal topological space and A is a closed
subset of X, then

IA-Ind(A) 6 I - Ind(X).

Proof. Obviously, if I - Ind(X) = −1 or I - Ind(X) = ∞, then the inequality
holds. We suppose that the inequality holds for all integers m < k and we
shall prove it for k. Let I - Ind(X) = k, FA be a closed subset of A and VA
be an open subset of A such that FA ⊆ VA. Since A is a closed subset of X,
FA is also a closed subset of X. Also, there exists an open set V in X such
that VA = V ∩ A. Since I - Ind(X) = k, there exists a ∗-open set U in X with
FA ⊆ U ⊆ V and IBd∗

X
(U)-Ind(Bd

∗
X(U)) 6 k − 1.

We consider the ∗-open set UA = U ∩ A in A. Then FA ⊆ U ∩ A ⊆ V ∩ A
or equivalently, FA ⊆ UA ⊆ VA. We shall prove that

(IA)Bd∗
A
(UA)-Ind(Bd

∗
A(UA)) 6 k − 1.

By Lemma 3.3 it suffices to prove that

IBd∗
A
(UA)-Ind(Bd

∗
A(UA)) 6 k − 1.

Since Bd∗A(UA) is a closed subset of Bd
∗
X(U), by inductive hypothesis we

have that

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 428



Small and large inductive dimension for ideal topological spaces

(IBd∗
X
(U))Bd∗

A
(UA)-Ind(Bd

∗
A(UA)) 6 k − 1.

Moreover, applying Lemma 3.3 we have that

(IBd∗
X
(U))Bd∗

A
(UA) = IBd∗A(UA).

Thus, IBd∗
A
(UA)-Ind(Bd

∗
A(UA)) 6 k − 1 and so IA-Ind(A) 6 k. �

The assumption that A is a closed subset of X cannot be dropped and the
following example justifies this.

Example 3.15. Let X = {a, b, c, d, e, f} with the topology

τ = {∅, {f}, {a, f}, {b, f}, {a, b, f}, {a, b, c, f}, {a, b, d, f}, {a, b, c, d, f}, X}.

If we consider the ideal I = {∅, {e}}, then I - Ind(X) = 0. However, if we
consider the set A = {a, b, c, d}, which is not closed in X, then IA-Ind(A) = 1.

Proposition 3.16. For any ideal topological space (X, τ, I) we have that

I - Ind(X) 6 min{Ind(X), Ind∗(X)}.

Proof. It is similar to Proposition 3.5. �

The following results characterize the ideal large inductive dimension in
various classes of spaces.

Proposition 3.17. An ideal topological space (X, τ, I), where (X, τ) is a nor-
mal space, satisfies the inequality I - Ind(X) 6 k, where k ∈ {0, 1, . . .}, if and
only if for every pair A, B of disjoint closed subsets of X, there exists a ∗-
partition L between A and B such that IL-Ind(L) 6 k − 1.

Proof. It is similar to the proof of Proposition 3.7. �

Corollary 3.18. A non-empty ideal topological space X is ideal zero dimen-
sional (with respect to the ideal large inductive dimension), where (X, τ) is a
normal space, if and only if for every disjoint closed subsets A and B of X, the
empty set is a ∗-partition between A and B.

Proof. It follows by Proposition 3.17. �

Proposition 3.19. If (X, τ, I) is an ideal topological space, where (X, τ) is a
normal space, and k ∈ {0, 1, . . .}, then the following statements are equivalent:

(1) I - Ind(X) 6 k,
(2) for every open neighborhood V of a closed set E of X, there exists a

∗-open set U and a ∗-closed set F in X such that

E ⊆ U ⊆ F ⊆ V and IF \U -Ind(F \ U) 6 k − 1,
(3) for every open neighborhood V of a closed set E of X, there exists a

∗-open set U in X with

E ⊆ U ⊆ Cl∗X(U) ⊆ V and IBd∗
X
(U)-Ind(Bd

∗
X(U)) 6 k − 1.

Proof. It is similar to the proof of Proposition 3.9. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 429



F. Sereti

The following result presents the behavior of the ideal large inductive di-
mension of the topological sum.

Theorem 3.20. Let {Xs : s ∈ S} be a set of pairwise disjoint topological
spaces and

⊕
s∈S Xs be their topological sum. Let also for each s ∈ S, Is be an

ideal of Xs. Then I - Ind(
⊕

s∈S Xs) 6 k, where I is the ideal of Lemma 3.10,
if and only if Is-Ind(Xs) 6 k for each s ∈ S.

Proof. It is similar to the proof of Theorem 3.11. �

4. Additional Results on Dimensions for Ideal Topological
Spaces

In what follows, properties of the ideal small and ideal large inductive di-
mension, using different ideals on the underlying sets, are studied. Moreover,
relations between the dimensions I - ind and I - Ind and the ideal covering di-
mension I - dim are investigated.

Proposition 4.1. Let (X, τ) be a topological space and I1, I2 be two ideals on
X.

(1) If I1 ⊆ I2, then I2-Ind(X) 6 I1-Ind(X).
(2) If I1 ⊆ I2, then I2-ind(X) 6 I1-ind(X).

Proof. For the simplicity of the writing, we denote by ∗1 and ∗2 the ∗-open set
referred to the ideal topological spaces (X, τ, I1) and (X, τ, I2), respectively.

(1) We suppose that the inequality holds for all integers m < k and we shall
prove it for k. Let I1-Ind(X) = k. We shall prove that I2-Ind(X) 6 k. Let F
and V be a closed and an open set in X, respectively, such that F ⊆ V . Since
I1-Ind(X) = k, there exists a ∗1-open set U in X such that F ⊆ U ⊆ V and

(I1)Bd∗1
X

(U)-Ind(Bd
∗1
X
(U)) 6 k − 1.

Since I1 ⊆ I2, the set U is also a ∗2-open set with Bd
∗2
X
(U) ⊆ Bd∗1

X
(U) and

thus, applying Lemma 3.3 and Theorem 3.4, we have that

(I1)Bd∗2
X

(U)-Ind(Bd
∗2
X
(U)) 6 k − 1.

Also, (I1)Bd∗2
X

(U) ⊆ (I2)Bd∗2
X

(U). Therefore, by inductive hypothesis, we

have that (I2)Bd∗2
X

(U)-Ind(Bd
∗2
X
(U)) 6 k − 1 and thus, I2-Ind(X) 6 k.

(2) It is similar to (1). �

Corollary 4.2. Let (X, τ) be a topological space and I1, I2 be two ideals on
X. Then

(1) max{I1-Ind(X), I2-Ind(X)} 6 I1 ∩ I2-Ind(X).
(2) max{I1-ind(X), I2-ind(X)} 6 I1 ∩ I2-ind(X).

Corollary 4.3. Let (X, τ) be a topological space, A, B subsets of X and I1 =
P(A), I2 = P(B) and I3 = P(A ∪ B) be three ideals on X. Then

(1) I3-Ind(X) 6 min{I1-Ind(X), I2-Ind(X)}.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 430



Small and large inductive dimension for ideal topological spaces

(2) I3-ind(X) 6 min{I1-ind(X), I2-ind(X)}.

Proposition 4.4. For any ideal topological space (X, τ, I), where (X, τ) is a
T1-space, we have that

I - ind(X) 6 I - Ind(X).

Proof. Let (X, τ, I) be an ideal topological space for which (X, τ) is a T1-space.
We suppose that the inequality holds for all integers m < k and we shall prove
it for k. Let I - Ind(X) = k. We shall prove that I - ind(X) 6 k. Let x ∈ X
and V be an open subset of X with x ∈ V . Then the set {x} is a closed subset
of X. Since I - Ind(X) 6 k, for the pair ({x}, V ) there exists a ∗-open subset
U of X with x ∈ U ⊆ V and

IBd∗
X
(U)-Ind(Bd

∗
X(U)) 6 k − 1.

Therefore, by inductive hypothesis, we have that IBd∗
X
(U)-ind(Bd

∗
X(U)) 6

k − 1 and thus, I - ind(X) 6 k. �

Example 4.5. We consider the space X = {a, b, c, d, e} with the topology

τ = {∅, {a}, {b}, {a, b}, {a, b, c}, X}

and the ideal I = {∅, {d}}. Then we observe that the space X is not T1,
I - ind(X) = 2 and I - Ind(X) = 0.

Especially, for the ideal small inductive dimension of X, we have the follow-
ing statements:

– for the element x = a and every open subset V of X with a ∈ V ,
there exists the ∗-open subset U = {a} of X such that a ∈ U ⊆ V and
IBd∗

X
({a})-ind(Bd

∗
X({a})) = 1,

– for the element x = b and every open subset V of X with b ∈ V , there
exists the ∗-open subset U = {b} of X such that a ∈ U ⊆ V and
IBd∗

X
({b})-ind(Bd

∗
X({b})) = 1,

– for the element x = c and every open subset V of X with c ∈ V , there
exists the ∗-open subset U = {a, b, c} of X such that c ∈ U ⊆ V and
IBd∗

X
({a,b,c})-ind(Bd

∗
X({a, b, c})) = 0,

– for the element x = d and every open subset V of X with d ∈ V ,
there exists the ∗-open subset U = X of X such that d ∈ U ⊆ V and
IBd∗

X
(X)-ind(Bd

∗
X(X)) = −1 and

– for the element x = e and every open subset V of X with e ∈ V ,
there exists the ∗-open subset U = X of X such that e ∈ U ⊆ V and
IBd∗

X
(X)-ind(Bd

∗
X(X)) = −1.

Therefore, I - ind(X) = 2.
Moreover, for the ideal large inductive dimension of X we observe that for

every pair (F, V ) of subsets of X, where F is closed, V is open and F ⊆ V ,
there exists the ∗-open subset U = X of X such that F ⊆ U ⊆ V and IBd∗

X
(U)-

Ind(Bd∗X(U)) = −1. Therefore, I - Ind(X) = 0.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 431



F. Sereti

For the rest of the paper, we remind that for a topological space (X, τ) a
non-empty family c of open subsets of X is called an open cover if the union of
all elements of c is X. A family r of subsets of X is said to be a refinement of
a family c of subsets of X if each element of r is contained in an element of c.

Especially, in what follows, for an ideal topological space (X, τ, I) a non
empty family c of open sets (respectively, of ∗-open sets) will be said a τ-cover
(respectively, a τ∗-cover) of X if the union of all elements of c is X.

The order of a family r of subsets of a topological space X is defined as
follows:

(1) ord(r) = −1, if r consists of the empty set only.
(2) ord(r) = k, where k ∈ {0, 1, . . .}, if the intersection of any k+2 distinct

elements of r is empty and there exist k+1 distinct elements of r whose
intersection is not empty.

(3) ord(r) = ∞ if for every k ∈ {1, 2, . . .} there exist k distinct elements of
r whose intersection is not empty.

Definition 4.6 ([17]). The ideal covering dimension, denoted by I - dim, is
defined as follows:

(i) I - dim(X) = −1 if and only if X = ∅.
(ii) I - dim(X) 6 k, where k ∈ {0, 1, . . .}, if for every finite τ-cover c of X

there exists a finite τ∗-cover r of X, which is a refinement of c with
ord(r) 6 k.

(iii) I - dim(X) = k, where k ∈ {0, 1, . . .}, if I - dim(X) 6 k and I - dim(X) 

k − 1.

(iv) I - dim(X) = ∞, if there does not exist any k ∈ {−1, 0, 1, 2, . . .} for
which I - dim(X) 6 k is true.

Proposition 4.7. Let (X, τ, I) be an ideal topological space, where (X, τ) is a
normal space. If I - dim(X) = 0, then I - Ind(X) = 0.

Proof. Let I - dim(X) = 0 and E, F disjoint closed subsets of X. Then the
family

c = {X \ E, X \ F}

is a τ-cover of X. Since I - dim(X) = 0, there exists a finite τ∗-cover r of X,
which refines c and has order less than or equal to 0. Thus, every point of X
is contained in a member of r, a member of r is disjoint from at least one of E
and F , and any two members of r are disjoint. Hence,

U =
⋃

{G ∈ r : G ∩ E 6= ∅} and V =
⋃

{G ∈ r : G ∩ E = ∅}

are disjoint ∗-open sets of X with E ⊆ U, F ⊆ V and X = U ∪ V . Thus, ∅ is
a ∗-partition between E and F with I∅-Ind(∅) = −1. Therefore, by Corollary
3.18, I - Ind(X) = 0. �

Corollary 4.8. Let (X, τ, I) be an ideal topological space, where (X, τ) is a
normal space. If I - dim(X) = 0, then I - ind(X) = 0.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 432



Small and large inductive dimension for ideal topological spaces

Proof. Since I - dim(X) = 0, by Proposition 4.7 we have that I - Ind(X) = 0
and thus, by Proposition 4.4, we have the desired result. �

Example 4.9. We consider the space X = {a, b, c} with the topology

τ = {∅, {a}, {b}, {a, b}, {b, c}, X}

and the ideal I = {∅, {b}}. Then I - ind(X) = I - Ind(X) = I - dim(X) = 0.

In general, we observe that the ideal topological dimensions of the types ind,
Ind and dim are different and the following examples prove this claim.

Example 4.10.

(1) We consider the space X = {a, b, c, d, e} with the topology

τ = {∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}, X}

and the ideal I = {∅, {c}}. Then I - ind(X) = 2 and I - dim(X) = 0.
(2) We consider the space X = {a, b, c, d} with the topology generated by

the family

β = {∅, {a}, {a, b}, {a, c}, {a, d}}

and the ideal I = {∅, {b}}. Then I - Ind(X) = 1 and I - dim(X) = 2.

However, under some conditions for topological spaces and ideals we can
obtain the following relation between I - ind, I - Ind and I - dim.

Proposition 4.11. Let (X, τ, I) be an ideal topological space. If I ⊆ τc and
(X, τ) is a separable metric space, then

I - ind(X) = I - Ind(X) = I - dim(X).

Proof. By Remark 2.7, Proposition 2.9 and [17], we have that ind(X) = ind
∗
(X),

Ind(X) = Ind∗(X) and dim(X) = dim∗(X). Also, based on the definitions of
the ideal topological spaces and the fact that τ = τ∗, we have that

I - ind(X) = ind(X), I - Ind(X) = Ind(X) and I - dim(X) = dim(X).

Finally, the desired relation follows from the fact that ind(X) = Ind(X) =
dim(X), whenever (X, τ) is a separable metric space. �

Acknowledgements. The author would like to thank the reviewer for the

careful reading of the paper and the useful comments.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 433



F. Sereti

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