EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 1344-1347
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Short Introduction to Similarity Via Ideals

Asli Guldurdek

College of Engineering and Technology, American University of the Middle East, Egaila
54200, Kuwait

Abstract. In this work, we introduce the notion of similarity between topologies τ1 and τ2, on a
set X via ideals. Then, we give some characterizations regarding this kind of similarity by using
∗−dense, and I−dense subsets. We also examine the preservation of similarity with respect to the
topologies τ∗1 and τ∗2 .

2020 Mathematics Subject Classifications: 54A05, 54A10, 54D80
Key Words and Phrases: Ideal topological space, Similarity

1. Introduction and Preliminaries

The idea of adding the notion of ideal into the topological spaces started with the works
of Kuratowski [6], and Vaidyanathaswamy [7]. After that the notion of ideal topological
space and applications have been examined deeply.

An ideal I on a set X is a nonempty collection of subsets of X, which satisfies the
following conditions:

i. If A ∈ I, and B ⊂ A, then B ∈ I

ii. If A,B ∈ I, then A ∪B ∈ I.

We denote a topological space (X, τ) with an ideal I defined on X by (X, τ, I). An
ideal I on (X, τ) is said to be τ -codense if I ∩ τ = {∅}.

On the other hand, there are many papers devoted to constructing new topologies via
ideals. To do that, firstly an operator called the local function is invented. Then by using
this, one can get a Kuratowski closure operator.

Definition 1 ([6]). Let (X, τ) be a topological space, and I be an ideal on X. Then the
local function A∗(I, τ) of A ⊂ X is defined as following:

A∗(I, τ) = {x ∈ X | U ∩A /∈ I for every U ∈ τ(x)}

where τ(x) = {U ∈ τ | x ∈ U}.

DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4465

Email address: asli.guldurdek@aum.edu.kw (Asli Guldurdek)

https://www.ejpam.com 1344 © 2022 EJPAM All rights reserved.



A. Guldurdek / Eur. J. Pure Appl. Math, 15 (3) (2022), 1344-1347 1345

One can see that, (.)∗ : P(X) → P(X) satisfies the conditions to make c∗(A) =
A ∪A∗(I, τ) a Kuratowski closure operator.

Definition 2 ([5]). Let (X, τ) be a topological space, and I be an ideal on X. Since
c∗(A) = A∪A∗(I, τ) is a Kuratowski closure operator, generates a topology τ∗(I, τ) on X.
If there is no chance of confusion this topological space is denoted as (X, τ∗).

Now we recall some definitions in ideal topological spaces, which are crucial in our work.

Definition 3. Let (X, τ, I) be an ideal topological space and A be a subset of X. We say
that A is

i. ∗-dense [4] if c∗A = X,

ii. I-dense [3] if A∗(I, τ) = X.

It is easy to show that τ ⊂ τ∗. Also note that, if I = {∅} then τ = τ∗, and if I = P(X)
then A∗(P(X), τ) = ∅ which implies τ∗ = P(X).

Carrying general topological notions into the ideal topological spaces is a very fruit-
ful, and generalizing process. To this end we turned our attention to similarity between
the topologies defined on the same set. This topic is introduced in [2]. According to
Bartoszewicz and et al. (X, τ1) and (X, τ2) are similar if the families of sets which have
nonempty interior with respect to τ1 and τ2 coincide. Similarity between topological spaces
is denoted by τ1 ∼ τ2. In [2], besides some other characterizations, it is shown that two
topologies are similar if and only if the families of dense subsets coincide, or τ1 \ {∅} and
τ2 \ {∅} are mutually coinitial [1]. That is for all U ∈ τ1 \ {∅} there exists V ∈ τ2 \ {∅}
such that V ⊂ U and for all U ∈ τ2 \ {∅} there exists V ∈ τ1 \ {∅} such that V ⊂ U .

In this work we first define similarity with respect to an ideal, and then give some
characterizations. Throughout this work, (X, τ), ciA, and c∗iA will denote the topological
space, closure in τi, and in τ∗i , respectively where i = 1, 2 .

2. Main Results

Definition 4. Let X be a set, τ1, τ2 be two given topologies, and I be an ideal on X. We
say that τ1 and τ2 are similar with respect to I or I-similar and denote by τ1 ∼I τ2 if, for
every nonempty U ∈ τ1, there exists a nonempty V ∈ τ2 such that V \U ∈ I, and for every
U ∈ τ2, there exists a nonempty V ∈ τ1 such that V \ U ∈ I.

It is clear that if τ1 and τ2 are similar topologies, then they are similar with respect
to any ideal I. On the other hand the following example shows that ideal similarity does
not imply similarity between topologies.

Example 1. Let X = {a, b, c}, τ1 = {∅, X, {a, b}}, τ2 = {∅, X, {b, c}}, and I = {∅, {a}, {c},
{a, c}}. τ1 and τ2 are I-similar, but not similar.

Also, if I = {∅}, then similarity is equivalent to I-similarity. What is more, if τ1 and
τ2 are I-similar topologies and J is an ideal with I ⊂ J, then τ1 and τ2 are also J-similar.



A. Guldurdek / Eur. J. Pure Appl. Math, 15 (3) (2022), 1344-1347 1346

Theorem 1. Let X be a set, τ1, τ2 topologies on X, and I be an ideal on X. Then τ1
and τ2 are similar with respect to I if and only if ∗-dense subsets coincide.

Proof. Let τ1 ∼I τ2, a subset A ⊂ X with c∗1A = X, and c∗2A ̸= X be given. By
definition, there exists an element x in X, so that x /∈ A and x /∈ A∗(I, τ2). Hence there
exists U ∈ τ2(x) so that U ∩A ∈ I. Together with this, c∗1A = X implies x ∈ A∗(I, τ1). By
I-similarity there exists a set V ∈ τ1 so that V \ U ∈ I. Note also that A ∩ V ̸= ∅. If we
consider A ∩ V , we see that A ∩ V ∈ I since A ∩ V ⊂ (U ∩A) ∪ (V ∩ (X \ U)). However,
that contradicts the fact that c∗1A = X.

On the other hand, let U ∈ τ1 \ {∅}, and suppose V \ U /∈ I for every V ∈ τ2 \ {∅}.
So, X \ U is ∗-dense with respect to τ2. By hypothesis X \ U is also ∗-dense with respect
to τ1. That is (X \ U) ∪ (X \ U)∗(I, τ1) = X, and this implies U ⊂ (X \ U)∗(I, τ1). As a
result we have the contradiction: U ∩ (X \ U) /∈ I.

Corollary 1. Let X be a set, τ1, τ2 topologies on X, and I be an ideal on X. Then τ∗1
and τ∗2 are similar if and only if τ1 and τ2 are I-similar.

Proof. By previous theorem, τ1 and τ2 are I-similar if and only if ∗-dense subsets
coincide, and by [2], Theorem 2.2, this is true if and only if τ∗1 and τ∗2 are similar topologies.

We have another characterization for I-similarity.

Theorem 2. Let X be a set, τ1, τ2 topologies on X, and I be an ideal on X. Then τ1
and τ2 are similar with respect to I if and only if I-dense subsets coincide.

Proof. Let τ1 ∼I τ2, A
∗(I, τ1) = X, and A∗(I, τ2) ̸= X. Then, there exists an element x

in X so that x /∈ A∗(I, τ2). That is, for a set U ∈ τ2(x), we have U ∩A ∈ I. By I-similarity
there exists a subset V ∈ τ1 \ {∅} such that V \ U ∈ I which implies V ∩ A ∈ I. That
contradicts the fact that x ∈ A∗(I, τ1) = X.

Let this time the families of I-dense subsets coincide, and U be a set belonging to
τ1 \ {∅}. Suppose V \ U /∈ I for every V ∈ τ2 \ {∅}. Then V \ U ̸= ∅, and hence X \ U is
I-dense in τ2. By hypothesis, X \ U is also I-dense in τ1, which brings the contradiction
that U ∩ (X \ U) /∈ I.

Lemma 1. Let X be a set, τ1, τ2 topologies on X, and I be an ideal on X. If τ1 and τ2 are
similar with respect to I, and I is a τi-codense ideal for i = 1, 2 then I is also τj-codense
ideal for j = 3− i.

Proof. Without loss of generality, assume I be a τ1-codense ideal, and let U ∈ I∩τ2\{∅}.
By I-similarity, there exists a set V ∈ τ1 \ {∅} so that V \ U ∈ I. However these imply
V ∈ I, which is impossible since I ∩ τ1 = {∅}.

Now, we examine the similarity between τ∗1 and τ∗2 . One can easily show that, if τ1
and τ2 are similar topologies, then τ∗1 , and τ∗2 are also similar. On the other hand the
converse is not true, as the following example shows:



REFERENCES 1347

Example 2. Let us reconsider the Example 1. The topologies τ∗1 , and τ∗2 satisfy:

τ∗1 = {X, ∅, {b}, {a, b}, {b, c}} = τ∗2 ,

are the same. However, as we mentioned, τ1, and τ2 are not similar.

Note that, the ideal of the previous example is codense with respect to both topolo-
gies, but this is not enough to have similarity between τ1, and τ2. This motivates the
following question:

Are there any conditions can be added to an ideal I for carrying similarity between τ∗1
and τ∗2 to the case of τ1 and τ2.

3. Conclusion

In this work the notion of similarity between topological spaces, is blended with ideals
on topological spaces. It is proved that, we can deduce the similarity with respect to an
ideal I, under the condition of coinciding I-dense subsets, and the similarity between τ∗1 ,
and τ∗2 topologies is equivalent the I-similarity between τ1 and τ2 topologies. However,
the question asking if there are any conditions can be added to an ideal I for carrying
similarity between τ∗1 and τ∗2 to the spaces τ1 and τ2 is still open for a possible future
work.

Acknowledgements

The author thanks the referees for their useful suggestions which were very helpful to
improve this article.

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