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International Journal of Advances in Applied Sciences (IJAAS)
Vol. 11, No. 4, December 2022, pp. 367~372
ISSN: 2252-8814, DOI: 10.11591/ijaas.v11.i4.pp367-372 367
Journal homepage: http://ijaas.iaescore.com
Interior topology: A new approach in topology
Amer Himza Almyaly
Department of Mathematics and Computer Applications College of Science, AL-Muthanna University, AL-Muthanna, Iraq
Article Info
ABSTRACT
Article history:
Received Aug 21, 2022
Revised Oct 25, 2022
Accepted Nov 3, 2022
This paper defined a new type of topology known as Interior topology. This
work falls among the types of topology (such as general topology, supra
topology, generalized topology, and filter) that are motivated by real-world
concepts such as the orbits of planets around the sun, electron orbits around
the nucleus, and so on. This form of topology is self-contained. The primary
objective of this study is to respond to the question “Is general topology
capable of producing Interior topology?”. Finally, we define the base for
Interior topology which is called i-base.
Keywords:
Filter
Generalized topology
Supra topology
This is an open access article under the CC BY-SA license.
Corresponding Author:
Amer Himza Almyaly
Department of Mathematics and Computer Applications College of Science, AL-Muthanna University
87QQ+2W6, Samawah, AL-Muthanna, Iraq
Email: amerhimzi@mu.edu.iq
1. INTRODUCTION
Numerous researchers have introduced new topological structures via either the topological
elements (open sets) or the topology itself (topology definition), such as Kelly [1] study of two topologies
determine for the same set named Bitopology and introduced various separation properties into topological
spaces, and obtained generalizations of some important classical results. Chang [2] introduced the definition
of fuzzy topological spaces and extended straightforwardly some concepts of crisp topological spaces to
fuzzy topological spaces. Then researchers [3], [4] introduced the concept of fuzzy Bitopological space and
defined the compactness of fuzzy topological space and the continuity, closeness, and openness of mapping
on the associated supra-fuzzy topological space. Shapir [5] defined soft topology on soft sets, and then Riza
et al. [6], defined N-soft topology on N-soft sets, which is an extension of soft topology. Tarizadeh [7]
defined flat topology in terms of the ring's prime spectrum. A flat topology is the dual of the Zariski topology
[8], [9] thus Zariski topology of a ring is a topology on the set of prime ideals, known as the ring spectrum.
Its closed sets are , where is any ideal in and is the set of prime ideals containing , and many
others who introduced new topological structures.
These structures were influenced by an idea, a relationship, the outside world, or natural
phenomena. They may or may not be real, but they will lay the framework for establishing the correct
scientific underpinnings, if we look up into the sky, a lot of thoughts come into our mind, and one of these
thoughts is "Can we divide the vast universe into portions under particular conditions (topology)?". The
purpose of this is to gain a deeper understanding and to broaden our perceptions. Here, we will discuss
mathematical ideas, as some may believe it is simply a discipline concerned with the language of numbers,
calculating, and symbols. Indeed, we cannot blame people for this belief, but it is a reality that everyone
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368
should be aware of mathematics is the language of the sciences, and it is the most precise approach for
proving hypotheses and claims across a wide variety of fields.
Nonetheless, we are attempting in our effort to introduce a novel idea of topology, which we have
dubbed (Interior topology). The structure of this topology corresponds to some phenomena around us, such
as the phenomenon of the movement of planets around the sun and the phenomenon of the movement of
electrons around the nucleus. This structure must be clearly defined and based on known topological
foundations. We will build a collection in which the infinite intersection is nonempty and attempt to
demonstrate its existence using examples and comparisons to various topologies such as general topology,
supra topology [10], generalized topology [11] and so filter [12].
The structure of Interior topology differs from that of other types of topologies as general topology
and supra topology. We can study their topological properties to get a better understanding of the things
around us. Finally, we shall have open sets known as i-open and so closed sets known as i-closed.
2. PRELIMINARIES
Definition 1: Let be a set. A topology on is a subclass of subsets of , called open sets (shr.
Open), such that satisfied the following:
a. and are open.
b. The intersection of finitely many open sets is open.
c. Any union (finite or infinite) of open set is open.
A topological space is set together with topology on .
Definition 2 [01]: Let be a set. A subclass is called supra topology on if:
a. .
b. is closed under an arbitrary union of elements of .
A supra topological space is set together with supra topology on
Remark 1: Every topological is supra topological space but the converse is not necessary as follows:
Example 1: Let be any infinite set and be a collection of all subsets which have more than one point,
then its obvious that is supra topology but it isn’t topology on .
Definition 3 [11]-[13]: Let be a set. A subclass is called generalized topology on if:
a. .
b. is closed under an arbitrary union of elements of .
A generalized topological space is set together with generalized topology on .
Remark 2: it's clear that every topological is generalized topological space but the converse is not necessary.
Remark 3: There exists no relation between generalized topological and supra-topological space.
Definition 4 [14]: Let be a nonempty set. A subclass is called a filter on if the following is
satisfied:
a. .
b. If then .
c. If and then .
Remark 4: There exists no relation between filter and topology.
3. INTERIOR TOPOLOGY
Definition 5: Let be a nonempty set. A subclass is called Interior topology on if the following
is satisfied:
a. .
b. is closed under an arbitrary union of elements of .
c. is closed under the arbitrary intersection of elements of .
An Interior topological space is set together with the Interior topology on .
Example 2: Let and then is satisfy the conditions
of Interior topology, thus is interior topological space.
Remark 5: Maybe does not belong to as follow:
Int J Adv Appl Sci ISSN: 2252-8814
Interior topology: A new approach in topology (Amer Himza Almyaly)
369
Example 3: Let and then is Interior topology on but
, thus is interior topological space.
Note 1: If we want to image the structure of definition 5, we can represent it as follows:
These orbits (Figure 1) suggest to us the shape of the orbits around the sun, and the region S originated from
the intersections of all these orbits. Now, we will discover the types of sets in this space.
Figure 1. Orbits
Definition 6: Let be an interior topological space, then the element is called i-open set (shr. i-
open) and the complement of is well called i-closed set (shr. i-closed), therefore we will reformulate
definition 5 as follow:
Definition 7: Let be a nonempty set and be a collection of subsets of (which is named i-open) then is
an Interior topology on if satisfy the following:
a. is not i-open.
b. The union of i-open sets is i-open set.
c. The intersection of i-open sets is i-open set.
Accordingly, we note the i-closed sets will satisfy dual of the above conditions as follow:
Proposition 1: Let be an interior topological space then the set of all i-closed satisfy the following:
a*. is not i-closed.
b*. The union of i- closed sets is i- closed set.
c*. The intersection of i- closed sets is i- closed set.
Proof:
a* For each i-open set , is i-closed, therefore for each is not exist in any i-closed set,
thus is impossible is i-closed set.
b* & c* is explain from complement from b & c.
Note 2: From above, we note the i-open set is impossible is i-closed set because it contains the intersection of
all i-open sets which is not contained in any i-closed set.
Definition 8: Let be an interior topological space and let , is called i-limit point of if
. The set of i-limit points of is called i-limit set of . The set of all i-limit points of all subsets in
is called the target set.
The target set satisfies some properties as follows:
Proposition 2: Let be an interior topological space then the following properties are equivalent:
a. is the target set.
b. , for each .
c. is a minimal i-open set.
Proof:
Let is the target set in , To prove , for each
Let , then from definition 4, , and at the same time every
element , for each is i-limit point of set , thus .
If , , then is contained in any i-open set therefore, is a minimal i-open set.
Let be the minimal i-open set and is the target set in then it's clear.
If such that , is i-limit point for some subset of and is contained in all i-open sets
for each is not minimal i-open contradiction, therefore, .
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Example 4: In real line , let
then are Interior topology on and the
target set is .
Example 5: In real line , let
then are Interior topology on and
the target set is .
Example 6: In real line , let then are Interior topology on and the
target set is .
Proposition 3: In any interior topological space, the target set is existing and unique.
Proof: Let be an interior topological space, since , for each in , is i-open set therefore, the
target set exists.
Let are two target sets in then
, for each in , therefore the target set is unique.
Note 1: In Figure 1, we may imagine the region as the target set.
3.1. Interior relative topology
Proposition 4: Let be an interior topological space and is the target set in and let such
that then the following collection is an Interior topology on .
Proof: Let then , thus
a.
.
b.
.
c. If such that and this is
a contradiction, therefore .
Now, we will define the interior relative topology.
Definition 9: The collection above is called interior relative topology on subset in Interior
topology and is called interior relative topological space.
Remark 6: Note if and as defined in proposition 4, then is not satisfied the conditions of
Interior topology because , therefore the relative topology is defined only on a subset which
intersection with the target set and this is one of the reasons why it is called Interior topology.
4. THE RELATION BETWEEN INTERIOR TOPOLOGY AND OTHER TOPOLOGIES
Here, we will prove there is no relation between Interior topology and topology, generalized
topology, supra topology, and filter. This section aims to prove the independence and existence of Interior
topology.
4.1. Topology and Interior topology
Every topology is impossible Interior topology and versa vice; we note that from Definitions 1 and
5, thus the topology has but the Interior topology hasn't , in addition, the condition of intersections mostly
does not come true in topology. But there is a relation between them we will discuss later in section 5.
4.2. Supra topology and Interior topology
The Supra topology maybe is Interior topology or not, i.e: i) Supra topology is Interior topology as
example 2; ii) Supra topology isn't Interior topology as remark 1. such that if we have supra topology which
is also topology then it is not topology as sub-section 4.1; and iii) Interior topology is not supra topology as
example 3.
4.3. The generalized topology and Interior topology
As sub-section 4.1, the generalized topology is impossible in Interior topology because it has and
versa vice.
4.4. The filter and Interior topology
The filter maybe Interior topology or not as follows:
a. The filter isn’t Interior topology as in the following example:
Example 7: In the real usual topological space, let be a collection of neighborhoods of number 0, then is
filtered on 0 but it isn’t Interior topology since the infinite intersection doesn’t belong to .
b. The filter is Interior topology as in example 2, the collection is satisfied with the conditions of the
filter and at the same time, it is Interior topology.
c. Interior topology isn’t filtered as the following example:
Int J Adv Appl Sci ISSN: 2252-8814
Interior topology: A new approach in topology (Amer Himza Almyaly)
371
Example 8: In example 4, is Interior topology but it isn’t filtered since
but
such that condition 3 of definition 4 isn't satisfied.
Now, we will discuss the necessary condition which makes the filter Interior topology.
Proposition 5: Let be a topological space and let filter in , if has the nonempty infinite
intersection of all it elements then is Interior topology.
Proof: Let be a filter in a topological space and let is the infinite intersection of all
elements of :
a. Let ℓ ℓ , but ℓℓ therefore ℓℓ by condition 3 of definition 4.
b. Let ℓ ℓ , but ℓℓℓ therefore ℓℓ by condition 3 of definition 4.
c. by condition 1 of definition 4.
From above then is the Interior topology on .
Remark 7: If we add the set , which is meaning the infinite intersection of all elements of , to filter then
can become not filter as the following example:
Example 9: In example 7, if we add , which is the infinite intersection of all elements of , to then
becomes not filter because the condition 3 of definition 4 such that for instance
but isn't neighborhood of .
5. I-BASE OF INTERIOR TOPOLOGY
Definition 10: Let be an interior topological space. The collection is called I-base for Interior
topology if every i-open is the intersection of elements of .
Example 10: In example 2, the I-base for is .
Example 11: In example 4, the I-base for is
.
Now, we will be discussing the following question: Is the topology generated by Interior topology?
Theorem 1: Let be a topological space and . Let be collection contains and all
elements of which contains then is I-base for Interior topology on target set .
Proof: Let be a topological space and . Let be collection contains and all elements
of which contains . Now, we suppose is a collection containing all intersections of elements of then:
a. Let ,
then
and this
means is the intersection of elements of therefore .
b. Let ,
then
(by [15]),
thus
are elements in that have , therefore, belong to , thus is the intersection of
elements of therefore .
c. From definition 4 and since the target set , then .
6. CONCLUSION
We defined an Interior topology in this work as one that is formed using an i-open set. However, we
investigated the structure's independence from the previously indicated structures. I-base also talks about
how the relationship between Interior topology and general topology is shown.
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BIOGRAPHIES OF AUTHORS
Amer Himza Almyaly is Associate Professor at Science Colloge AL-Muthana
University, Iraq. He Holds a PhD degree in Mathematics Science. He also holds a Higher
Diploma in Computer Science. His research areas are Topology, Fuzzy Topology, Fuzzy Sets
and Functional Analysis. Head of Department Mathematics and Computer Applications in
Science Colloge AL-Muthana University from 2019 to 2020. He can be contacted at email:
amerhimzi@mu.edu.iq.
