On Light Mapping and Certain Concepts by Using mXN-Open Sets

The aim of this paper is to present a weak form of m-light functions by using mXN-open set which is mN-light function, and to offer new concepts of disconnected spaces and totally disconnected spaces. The relation between them have been studied. Also, a new form of m-totally disconnected and inversely mtotally disconnected function have been defined, some examples and facts was submitted.


Introduction:
In (2016) Abass and Ali (1) introduced the definition of -light function, Humadi and Ali (2) presented the ̂-light function. Al Ghour and Samarah (3) defined -open set. In this research we defined the set -open set, we submitted a new type of functions by using -open sets, it is weaker than -light function and we named it light function. In (4) Carlos Carpintero, Jackeline Pacheco, Nimitha Rajesh, Ennis Rafael Rosas and S. Saranyasri defined -connected space, by the same manner -disconnected, -disconnection, -disconnected, -connected and -totally disconnected spaces have been defined, additionally, many types of functions in -structure spaces such as -totally disconnected, *totally disconnected, * * -totally disconnected, inversely -totally disconnected function have been introduced. In (5) Enas Ridha Ali, Raad Aziz Hussain introduced the definition of -hausdorff, and in the same way, -hausdorff has been defined. Also 1 -spaces and zero dimensionspaces have been provided. The relation between these concepts has been discussed. Moreover the relation between -homeomorphism functions (6) and the -light functions has been illustrated. Examples, theorems and some facts supported our study.

Main Results:
In this section, -totally disconnected, light functions and some spaces by using open sets have been presented. Definition 1 (7), (8) A subcollection of the power set ( ) of a non-empty set is called a minimal structure on if ∅, ∈ , the pair ( , ) is called -structure space (in short -space). Each element in is said to be -open set and its complement is said to be -closed set.

Definition 4 (1)
An -space is said to be -disconnected, if there are non-empty -open sets and in such that ⋃ = and ⋂ =∅, if is not mdisconnected space then it is called -connected space.

Definition 5
Let ( , ) be an -space and , are two nonempty -open subsets of , we call ⋃ to be -disconnection to , if ⋃ = and ⋂ =∅. In example 3 -{ } and { } where ∈ , are disconnection to .

Definition 6
An -space is -disconnected if we can find an -disconnection to it, if there is no such disconnected so is -connected space.

Proposition 1
An -space is -disconnected if and only if there is a non-empty -clopen subset in such that ≠ .

Proof
Suppose is a non-empty -clopen subset of such that ≠ . Let = , so is a subset of and ≠∅ (because ≠ , and ⋃ = , ⋂ =∅). Also is -clopen because is -clopen, therefore is -disconnected space. Conversely, if is -disconnected space, so there is an disconnection ⋃ to , hence = which implies is -closed subset of , therefore is -clopen subset of and ≠ since is nonempty subset of , and then is a non-empty -clopen subset of such that ≠ .

Proposition 2
An -space is -connected space if and only if ∅ and are the only -clopen set in .

Proof
If is an -connected space, and is a nonempty proper -clopen subset of , then is also -clopen subset of , and since ⋃ = , where ≠∅, therefore is -disconnected space and that is a contradiction, so ∅ and are the only -clopen set in . Conversely, suppose is -disconnected space, so there is disconnection ⋃ to , but L is -closed (since = ) which is a contradiction, therefore is -connected.

Definition 7
The -space ( , ) is called an -totall disconnected space. If for every pair of distinct points a and b in , there are two -open sets , such that ≠∅, ≠∅, ∈ , ∈ , ⋃ = and ⋂ =∅ .

Example 5
For any distinct points , in the discrete -space I-Every -connected space is -connected but the converse is not true, since if ( , ) is an connected space, and suppose it is -disconnected space then there is -disconnection ⋃ to , and then it is -disconnected (by Remark 2) which is a contradiction, hence is -connected.

Proof
If is an -disconnected subset of so there is -disconnection ⋃ to , and then there are -open sets and in such that = ⋂ and = ⋂ , therefore ⊆ ⋃ , ⋂ ≠∅, ⋂ ≠∅ and ⋂ ⋂ =∅. Conversely, since ⋂ and ⋂ are separate , so is -disconnected subset of .

Definition 8
The -closure for a subset of -space is the intersection of all -closed sets of which containing and it is denoted by -( ). And the -interior for a subset of -space is the union of all -open sets of which containing in and it is denoted by -( ).

Proposition 5
If is a subset of an The -continuous image of -connected set in is -connected set in .

Note 1
An -space is called is 1 -space if for each two distinct points , in there are two nonempty -open sets and such that containing but not and containing but not .

Definition 10
An -space is called is -Hausdorff space if for each distinct points , in there are two nonempty -open sets and in such that ∈ , ∈ and ⋂ =∅.

Example 11
Let ( , ) be the indiscrete -space, let , ∈ with ≠ , then we can find two -open sets and in such that = -{ } which containing but not , and = -{ } which containing but not , so ( , ) is

Remark 6
Every -totally disconnected space is -Hausdorff space, but the converse is not true, since if is -totally disconnected space then for each distinct points a, b in , we can find two -open sets , containing a, b respectively with ⋂ =∅ and ⋃ = , so is -Hausdorff space.

Example 12
Let (ℛ, ) be the usual -space, it is -Hausdorff space, but not -totally disconnected.

Remark 7
Every -Hausdorff space is -Hausdorff, but the converse is not true, since if is -Hausdorff space, so there are -open sets and in , such that ≠∅, ≠∅, and ∈ , ∈ , by Remark 2 is -Hausdorff.

Remark 8
Every -totally disconnected space is disconnected but the converse is not true, since if is -totally disconnected space, then for any two points , ∈ where ≠ we can find -open sets and in , with ≠∅, ≠∅, ⋂ =∅, and they containing a, b respectively such that ⋃ = , so is -disconnected and then -disconnected (by remark 4)).

Example 15
Let

Definition 11
The -function : ( , ) ⟶ ( , ) is called -totally disconnected if the image of each totally disconnected set in is -totally disconnected in .

Definition 12
The -function : ( , ) ⟶ ( , ) is called * -totally disconnected if the image of each -totally disconnected set in is totally disconnected in .

Definition 13
The -function :( , ) ⟶ ( , ) is called * * -totally disconnected if the image of each -totally disconnected set in is -totally disconnected in . The following Example satisfying Definitions 11, 12 and 13.

Definition 14
The surjective -function : ( , ) ⟶ ( , ) is called -light function if the inverse image of any ∈ is -totally disconnected set in .

Remark 10
Every -light function is -light function, but the converse is not true, since if :(X, m X ) ⟶(Y, m Y ) is -light function, then −1 ( ) is -totally disconnected for any in , then it is -totally disconnected set in (by Remark 2), so is light function.

Remark 11
Every -homeomorphism function is -light function, but the converse is not true, since if :(X, m X ) ⟶(Y, m Y ) is -homeomorphism function, then for any in there is a unique in where ( )= (since is bijective), so −1 ( )={ } which is -totally disconnected, so {a} is - totally disconnected (by Remark 2), and then is -light.

Definition 15
A surjective -function :( , ) ⟶( , ) is called inversely -totally disconnected function if the inverse image of any -totally disconnected set in is -totally disconnected set in .

Example 20
The identity -function :( , is a finite set is inversely -totally disconnected function.

Proposition 8
Every inversely -totally disconnected function is -light function.

Proof
Let :( , ) ⟶( , ) be inversely -totally disconnected function and ∈ , since is surjective -function (since it is inversely totally disconnected) and −1 ({ }) is -totally disconnected set in , where {b} is -totally disconnected set in which implies is -light function.

Definition 16
The -space ( , ) is called a zero dimension -space if it has a base of -clopen sets.

Lemma 2
Every zero dimension metric -space is totally disconnected space.

Proof
Let be a zero dimension metric -space and , are points in with ≠ , then is -Hausdorff space and since it is metric -space, then has a neighbourhood with ∉ , then there exists a basic -open set which is also -closed set in (since is zero dimensional -space) and then is -clopen set (by Remark 2 and since the complement of -open set is -closed set), where ∈ ⊆ , and is -clopen set in such that ∈ , = ⋃ and ⋂ =∅, so is -totally disconnected space.

Proposition 13
Let , be metric -spaces and :( , ) ⟶ ( , ) be a surjective -function where is compact space, then is -light function if the inverse image for each ∈ is a zero dimension a subspace of .

Proof
Let ∈ , so −1 ( ) is zero dimension metricsubspace of (since metric is hereditary property), so it is -totally disconnected subspace of (by lemma ) and so that is -light function. New subjects and future work.

Definition 17 (12)
A subset of -space is said to be -gclosed if for each -open set with ⊆ , then -cl ( ) ⊆ .

Definition 18
A subset of -space is said to be -g-open if ⊆ -Int ( ) for each -closed set with ⊆ .

Definition 19
A subset of -space is said to be -gopen set if for each ∈ , there exists -g-open set containing such that -is finite. The complement of -g-open set is -g-closed set. There is a relation between Definition 19 and --open set as follows.

Remark 12
Every --open set is -g-open, but the converse is not true in general.

Example 21
The subset { } ∈ℛ in (ℛ,  If we use -g-open set instead of -open in this research, will we get approach results? Now we will use the previously presented set to define another type of -disconnected space, which is:-Definition 20 An -space is said to be -g-disconnected if it is union of two disjoint -g-open sets.

Question 3
What is the relation between --disconnected and -g-disconnected space? In a same way and by using -g-open set, new type of -light function have been defined, which is:-

Definition 21
A function from -space into -space is said to be -g-light if for every ∈ , −1 ( ) is -g-totally disconnected.

Question 4
What is the relation between -light and -glight function?

Remark 13
There is a definition in the topological space to Nadia Kadum Humadi (13), we can exploit it by using the definition of -g-open set.

Conclusions:
In this research, new spaces namely disconnected, -totally disconnected, -Hausdorff, 1 -spaces, have been defined and -light and inversely -totally disconnected functions have been introduced.