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Original Article | Open Access | Aust. J. Eng. Innov. Technol., 2025; 7(2), 160-168. | doi: 10.34104/ajpab.025.01600168

Preservation of Topological Properties under Some Maps

Suhaila Barekzai* Mail Img Orcid Img ,
Mustafa Danesh Mail Img Orcid Img ,
BiBi Amina Salimi Mail Img Orcid Img

Abstract

In this paper we investigate preservation of some topological properties during the weakly discontinuous, scatteredly continuous and point-wise discontinuous maps on topological spaces. In addition, we also investigate preservation of topological properties during such corresponding homeomorphisms. Furthermore we also establish some significant relationship among weakly discontinuous, scatteredly continuous and point-wise discontinuous maps. Scatteredly continuous map, weakly discontinuous map, point-wise discontinuous map, weak homeomorphism, scattered homeomorphism, point-wise homeomorphism. 

Introduction

The exploration of continuity and discontinuity within topology is essential for understanding the structural attributes of topological spaces. Over time, various forms of discontinuous mappings have been developed to extend traditional continuity concepts. Among these, weakly discontinuous, scatteredly continuous, and point-wise discontinuous functions have received considerable attention due to their relevance in functional analysis and general topology (Banakh & Bokalo, 2010; Arkhangelskii, 1978a). These mappings display distinctive preservation properties that are crucial for characterizing topological structures, holding significance in both theoretical and applied mathematics.

A function ├ f:X→Y┤ between topological spaces is referred to as weakly discontinuous if the set of discontinuity points is nowhere dense in Y for every non-empty subspace of X (Banakh & Bokalo, 2017). In contrast, a function is scatteredly continuous if it has at least one point of continuity in every non-empty subspace of X (Banakh & Bokalo, 2019). The notion of point - wise discontinuity further extends this concept by ensuring that each non-empty closed subspace includes a point of continuity (Bokalo & Kolos, 2011). These mappings, along with their corresponding homeomorphisms - such as weak, scattered, and point-wise homeomorphisms - have been studied for their role in preserving essential topological properties (Banakh, Bokalo, & Kolos, 2017; Bokalo & Kolos, 2019; Azadpour et al., 2023).

Numerous studies have examined how these mappings affect various topological invariants, including compactness, Lindelöfness, and hereditary properties (Bokalo & Kolos, 2011; Bokalo & Kolos, 2019). The preservation of these properties under different mappings has been a focal point in mathematical literature, particularly in relation to the classification and structural behavior of topological spaces (Preiss & Simon, 1974; Vinokurov, 1985). Understanding these dynamics is critical for broadening classical results to more generalized frameworks, especially in spaces exhibiting intricate continuity behaviors.

The preservation of topological properties under various types of discontinuous mappings has been a significant focus in mathematical research. Numerous studies have contributed to the understanding of weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps, as well as their implications for general topology. Arkhangelskii, (1978a) laid the groundwork for the classification of topological spaces and cardinal invariants, which are essential for analyzing the behavior of different types of mappings in topology. Expanding on this foundation, Arkhangelskii and Bokalo, (1992) investigated the tangency of topologies and the tangential properties of topological spaces, offering valuable insights into how certain properties are preserved or transformed under various mappings.

The study of scatteredly continuous functions was advanced by Banakh and Bokalo, (2010), who examined their role in maintaining key topological characteristics. Later, Banakh and Bokalo, (2017) extended their research to weakly discontinuous and resolvable functions, emphasizing their significance in general topology. Further exploration of functional generalizations of regular topological spaces by Banakh and Bokalo, (2019) reinforced the importance of discontinuous mappings in understanding the structural properties of topological spaces. Banakh, Bokalo, and Kolos, (2017) provided a comprehensive analysis of weakly discontinuous maps and weak homeomorphisms, demonstrating how these mappings preserve essential topological properties. Additionally, Bokalo and Kolos, (2011) examined the normality of spaces associated with scatteredly continuous maps, contributing to a broader understanding of their role in topological studies. Their concurrent research on operations involving discontinuous maps further explored the structural effects of these functions.

Bokalo and Kolos, (2019) also investigated the invariance of the Lindelöf number under discontinuous mappings, illustrating their impact on compactness -related properties. In the context of functional analysis, Preiss and Simon, (1974) studied weakly pseudocompact subspaces of Banach spaces, highlighting their relevance to topology. Furthermore, Vinokurov, (1985) explored the strong regularizability of discontinuous functions, identifying conditions under which discontinuous functions can exhibit regular behavior.

These contributions provide a theoretical foundation for understanding the preservation of topological properties under weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps. The present study builds upon this existing body of work by further examining the interconnections among these function classes and their implications for homeomorphisms and broader topological structures. This paper aims to investigate the preservation of topological properties under weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps. Furthermore, we examine how these properties extend to the corresponding homeomorphisms, offering new insights into the relationships between these function classes. The findings presented in this study contribute to the field of topology by providing a new perspective on generalized continuity and its ramifications.

Methodology

Prelimaneries

A map h: A→B between two topological spaces A and B is said to be   

 w-discontinuous, if the set D(h|_X) is nowhere dense in X for each non-empty subspace X of A. 

s-continuous, if the map h|_X has a point of continuity (i.e. the set C(h|_X)≠∅) for each non-empty subspace X of A. 

p-wise discontinuous, if the map h|_X has a point of continuity (i.e. the set C(h|_X)≠∅) for each non-empty closed subspace X of A. 

Where, D(h|_X) and C(h|_X) are sets of discontinuity and continuity points of the restriction map h|_X respectively.

A bijective map h:A→B is weakly homeomorphism (briefly, w-homeomorphism) if h and h^(-1) are weakly discontinuous maps. Similarly, map h is scatteredly homeomorphism (briefly, s- homeomorphism) if h and h^(-1) are scatteredly continuous. Recall some concepts from the literature [6] as follows.

Let M is family of subsets of topological space A. The smallest cardinal ((briefly, s-cardinal) |M| s.t. for every point a∈A and for each neighborhood (briefly, nbd) P⊂A of point a, there is a set M∈M s.t. a∈M⊂P is called network weight and denoted by nw(A). If X is subspace of A, then the s-cardinal κ s.t. every open cover C of X contain C_1 as a subcover of |C_1 |≤κ is called heredity Lindelof number and denoted by hl(A). If X has a subset U as a dense set s.t. |U|≤κ, then s-cardinal κ is called hereditary density and denoted by hd(A).

A property ρ of (regular) space is called:  

κ-summable: If D the collection of topological spcaes which persevere the property ρ and

|D|≤κ, then the topological sum ⊕_  D of family D has that property ρ. 

Closed (open)-hereditary: If A preserved property ρ, then each closed (open) or finite subspace of A has property ρ and denoted by c-hereditary. 

Projective: If h:A→B is surjective continuous function and A persevered property ρ, then

B=h(A) has property ρ and denoted by p-property. 

κ-additive: If A has a cover D by ≤κ many closed subspace which preserved property ρ, then A has property ρ, where κ is cardinality of A. 

σ-additive: If (regular) space A has property ρ, then it has a countable closed cover C which persevered property ρ.

Topological: If A persevered property ρ, provided A is homeomorphic to space B that has property ρ. 

Scatteredly-projective: Let h:A⟶B is a (scatteredly) continuous map from A to (regular) space B. If B persevered property ρ, then B is the image of space A with property ρ under function h. 

The s-ordinal β s.t. ¯D_(β+1) (h)=¯D_β (h) is called index of w-discontinuous map and denoted by wd(h). A function h is w-discontinuous if and only if ¯D_(wd(h)) (h)=∅. The s-ordinal β s.t. D_α (h)=∅ is called index of s-continuous and denoted by sc(h). A map h: A→B is s-continuous, if D_α (h)=∅.

In [6], the large pseudocharacter ϕ(A) of space A is the s.cardinal κ s.t. we can write every open subset V of A as union of family H having ≤κ many finite or closed subsets of A, i.e. V=⋃_ ▒‍ H, (ϕ(A)≤hl(A)) and the closed decomposition number ¯dec(h) of function h:A⟶B is the s.cardinality |D| of cover D of A by finite or closed subsets s.t. the map h|_D:D⟶B is continuous ∀D∈D. Furthermore, the map h:A⟶B is continuous iff ¯dec(h)=1.

Let A be any space, let C and D be the open and closed covers of A respectively. Consider the family B_α of the locally finite closed subsets of A. The s.cardinal κ s.t. each open cover C of A can be refined by closed cover D of A and D=⋃_(α<κ)▒‍ D_α of κ many locally finite collection B_α of closed subsets of A is called para-compact number of space A and denoted by par(A). Further, par(A)≤1 if and only if A is a para-compact space.

Let h:A→B be a w-discontinuous function then-

¯dec(h)≤|wd(h)|.ϕ(A). 

|wd(h)|.ϕ(A)≤hl(A), if A is regular. 

¯dec(h)≤ϕ(A), if A is para-compact. 

¯dec(h)≤ω, if A is perfectly para-compact. 

¯dec(h)≤max{par(A),ϕ(A)}. 

¯dec(h)≤hl(A), if A is regular. 

nw(B)≤nw(A) 

hl(B)≤hl(A) 

hd(B)≤max{hd(A),hl(A)} 

A property ρ of A is called:  

Preserved by w-homeomorphisms: Let h:A→B be a w-homeomorphism, then space A preserved property ρ if and only if B has property ρ. 

Closed+Open-additive: If A has property ρ then there is an open subset P of A s.t. P and the complement of P preserve property ρ. 

Local: If A preserved property ρ if and only if each point a∈A has a nbd P⊂A which has property ρ. 

Scattered: A space A∈ρ if and only if each closed subspace U≠∅ of A has a relatively open subset V≠∅ which preserve property ρ. 

If map h:A→B is weakly (scattered) homeomorphic perfectly para-compact, then  

nw(A)=nw(B) 

hl(A)=hl(B) 

hd(A)=hd(B) 

hd(A).hl(A)=hd(B).hd(A) 

dim(A)=dim(B) 

A is σ-compact if and only if B is σ-compact. 

In some other terms the following important result for w-discontinuous, s-continuous and p-wise discontinuous maps was established in [3,6] and [8]. We provide proof here for convenience.

Lemma 2.1  Let A be a topological space and γ≠0 ordinal. Let the transfinite sequence (F_β )_β≤γ of closed subsets of A s.t. F_γ=⌀, F_0=A and F_(β+1)⊂F_β for any β<γ and for any limit ordinal (briefly, l.ordinal) β≤γ, F_β=⋂_(α<β)▒‍ F_α. Then the identity function i:A⟶⊕_(β<γ) F_βF_(β+1) is

 w-discontinuous. 

Lemma 2.1  Let h is s-continuous from space A to a regular space B, then each X≠⌀ subspace of A which contains an open dense U⊂X s.t. the map h|_X:X⟶B is continuous for every u∈U. 

Theorem 2.2  Let h:A⟶B and g:B⟶C be s-continuous maps. If B is regular, then the composition goh:A⟶C is a s-continuous. 

Proof. Given h:A⟶B is a s-continuous , where B is regular. Let U be an arbitrary subspace of A. The subspace U contains an open dense subset V (by 1.2) s.t. the restriction map h|_U:U⟶B is continuous for every v∈V. Since given g:B⟶C is s-continuous, so we should find a continuity point b∈h(A) of restriction function g|_(h(U)):h(U)⟶C. Consider an arbitrary point a∈h^(-1) (b)∩V, the composition map goh:A⟶C is continuous at a∈A. 

Theorem 2.3  Let h:A⟶B be a w-discontinuous surjective map from a perfectly para-compact A to B. If A has c-heredity ω-summable p-property ρ, then B has that property, too. 

Proof. Given A is perfectly para-compact and preserved ω-summable property, therefore ¯dec(h)≤ω, i.e. we have a cover D of A by finite or closed subsets of A s.t. |D|=¯dec(h)≤ω and the function h|_D:D⟶B is continuous ∀D∈D.

Assume the topological sum of cover D as ⊕_  D, given A has c-heredity and ω-summable property, therefore ⊕_  D preserved the property ρ also. Now consider the surjective continuous function, 

H:⊕┬  D⟶B 

H(D,a)=h(a)    ,    ∀D∈D    and    a∈D

H is surjective continuous and ⊕_  C has p-property ρ, therefore B has that property. 

Theorem 2.4  Let h:A⟶B be a w-discontinuous surjective function from Hausdorff space A to B. If A has h-lindelof and σ-compact, then B has that property, too. 

Proof. Let us assume A has h-lindelof and σ-compact, to prove B has that property too. Let ¯D_β (h) be the subspace of A and C be the open cover of ¯D_β (h). Since A has h-lindelof, so we can find a subcover of cardinality |C_1 |≤κ. Given A is Hausdorff and every open cover of ¯D_β (h) has a finite subcover implies ¯D_β (h) is compact and closed.

Since every compact space is σ-compact implies ¯D_β (h) is σ-compact. Consider a t.sequence of closed subsets of A as (¯D_α (h))_(α<κ) s.t. D_κ=⌀, ¯D_0 (h)=A and ¯D_β (h)=⋂_(α<β)▒‍ ¯D(h|_(¯D_α (h))) for l.ordinal β≤κ and ∀α<κ. Since the set ¯D_α (h) is closed in A, so ¯D_α (h)¯D_(α+1) (h) is dense in ¯D_α (h) and the map 

h|_(¯D_α (h)¯D_(α+1) (h)):¯D(h)_α¯D_(α+1) (h)⟶B

is continuous for every α<κ. Since the property ρ is h-lindelof and σ-compact, so the space ¯D_α (h)¯D_(α+1) (h) preserved the property ρ. Hence B has the property ρ. 

Theorem 2.5  Let A and B be topological spaces and let A_1 and B_1 be closed subspaces of A and B respectively, if A and B are homeomorphic to B_1 and A_1 respectively, then A and B are 

w-homeomorphic. 

Proof. Let us assume that A is homeomorphic to B_1 and B is homeomorphic to A_1, where A_1 and B_1 are closed subspaces of A and B respectively. Let A_0=A and B_0=B, fix homeomorphisms h:A⟶B_1 and g:B⟶A_1, define subsets as h(A_m)=B_(m+1) and g(B_m)=A_(m+1) for m≥0. Since A_1⊂A and B_1⊂B implies A_2=g(B_1)⊂g(B)=A_1 and B_2=h(A_1)⊂h(A)=B_1 implies A_2⊂A_1 and B_2⊂B_1, continue this posses by mathematical induction, we get A_(m+1)⊂A_m and B_(m+1)⊂B_m for all m∈ω, where the sets A_m and B_m are closed in A and B respectively.

Now consider the sets A_∞=⋂_(m∈ω)▒‍ A_m and B_∞=⋂_(m∈ω)▒‍ B_m 

h(A_∞)=h(⋂_(m∈ω)▒‍ A_m)=⋂_(m∈ω)▒‍ B_m=B_∞

Therefore, the sets A_∞ and B_∞ are closed in A and B respectively. Define a bijective map H:A⟶B s.t. H(a)=h(a) for a∈A_∞∪⋃_(m∈ω)▒‍(A_2m-A_(2m+1)) and H(a)=g^(-1) (a) 

for a∈⋃_(m∈ω)▒‍(A_(2m+1)-A_(2m+2)). By lemma 1.1, the bijective map h:A⟶B is w-homeomorphism.

Corollary 2.2 Let A and B be topological spaces and let A_1 and B_1 be closed subspaces of A and B respectively, if A and B are homeomorphic to B_1 and A_1 respectively, then A and B are

 s-homeomorphic.

Theorem 2.6  Let h:A⟶B be a w-homeomorphism between topological spaces A and B, then there are covers {A_i:i∈I} and {B_i:i∈I} for topological spaces A and B respectively, by finite or closed subsets, for the set I of |I|≤¯dec(h).¯dec(h^(-1)), s.t. ∀i∈I the map h|_(A_i ) is homeomorphism of A_i onto B_i.  

Proof. Since h:A⟶B is bijective map, therefore h^(-1) is exist. By the definition of ¯dec(h), we have a cover {A_β:β<¯dec(h)} of A by finite or closed subsets s.t. ∀β<¯dec(h) the map h|_(A_β ) is continuous and there exist a cover {B_γ:γ<¯dec(h^(-1))} of B by finite or closed subsets for the map h^(-1):A⟶B s.t. ∀γ<¯dec(h^(-1)) the map h^(-1) |_(B_γ ) is continuous.

Let I=¯dec(h).¯dec(h^(-1)) and ∀i=(β,γ)∈I, let A_i=A_β∩h^(-1) (B_γ) and B_i=B_γ∩h(A_β). Hence the map h|_(A_i ) is homeomorphism of A_i onto B_i, for every i∈I. 

Corollary 2.3  Let h:A⟶B be a s-homeomorphism between regular spaces A and B. Then there are covers {A_i:i∈I} and {B_i:i∈I} for topological spaces A and B respectively, by finite or closed subsets, for the set I of cardinality 

|I|≤max{par(A),par(B),ϕ(A),ϕ(B)}

s.t. for each i∈I the map h|_(A_i ) is homeomorphism of A_i onto B_i. 

Theorem 2.7  Let h:A⟶B be a bijective map and ρ be κ-additive c-hereditary top-property, then A preserved property ρ iff B has property ρ, where κ=¯dec(h).¯dec(h^(-1)). 

Proof. Let us assume A preserved κ-additive, so A has a cover C by the closed subspaces which has property ρ, where |C|<κ_1=¯dec(h). Let C={C_α:α<¯dec(h)} be the cover of the space A by finite or closed subsets s.t. ∀α<¯dec(h) the map h|_(C_α ):C_α⟶B is continuous (by the definition of ¯dec(h)). Since the cover C has property ρ, so each closed or finite subset C_α, where α<¯dec(h) has property ρ (A has c-hereditary property) and ρ is topological, therefore B preserved the property ρ.

Conversely, for the function h^(-1):B⟶A, assume B preserved the property ρ, so B has a cover D by the closed subspaces which preserved ρ, where |D|<κ_2=¯dec(h^(-1)). Let D={D_β:β<¯dec(h^(-1))} be the cover of the B by finite or closed subsets s.t. ∀β<¯dec(h^(-1)) the map h^(-1) |_(D_α ):D_α⟶A is continuous (by the definition of ¯dec(h^(-1))). Since the cover D has property ρ, so each closed or finite subset D_β, where β<¯dec(h^(-1) has property ρ (B has c-hereditary property) and ρ is topological, therefore A has the property ρ.

Hence, A preserved property ρ iff B has property ρ, where κ=κ_1.κ_2=¯dec(h).¯dec(h^(-1)). 

Theorem 2.8  Let h:A⟶B be a w-homeomorphism and let ρ be κ-additive c-hereditary top-property. Then A has property ρ iff B preserved property ρ, where κ=hl(A).hl(B).ϕ(A).ϕ(B). 

Proof. Since h is w-homeomorphism, so by the w-discontinuity of maps h and h^(-1), we can write ¯dec(h)≤|wd(h)|.ϕ(A) and ¯dec(h^(-1))≤|wd(h^(-1))|.ϕ(B). Since wd(h)≤hl(A)^+, i.e. wd(h) can not exceed hl(A)^+, the successor cardinal of the hl(A) of space A, therefore ¯dec(h)≤hl(A).ϕ(A) and ¯dec(h^(-1))≤hl(B).ϕ(B). Hence, κ=¯dec(h).¯dec(h^(-1))≤hl(A).ϕ(A).hl(B).ϕ(B).

Finally, if h:A⟶B is a w-homeomorphism between A and B and ρ is κ-additive c-hereditary top-property, where κ=hl(A).ϕ(A).hl(B).(A).ϕ(B), then A preserved property ρ iff B preserved property ρ, too (by 1.10). 

Theorem 2.9  Let h:A⟶B be a w-homeomorphism between perfectly para-compact spaces A and B and let ρ be ω-additive c-hereditary property ρ. If A preserved property ρ, then B has property ρ, too. 

Proof. Let us assume that A has property ρ, so by the definition of ω-additive A has a cover C by closed subspace which preserved property ρ s.t. |C|≤ω. Since A is perfectly para-compact space and h is w-discontinuous map, so ¯dec(h)≤ω. Let C={C_β:β<ω} s.t. |C|=¯dec(h)≤ω and the map h|_(C_β ):C_β⟶B is continuous for each β<ω. Property ρ is c-hereditary and ω-additive implies C_β preserved property ρ for each β<ω.

Define a w-discontinuous map as h^(-1):B⟶A, since B is perfectly para-compact, so ¯dec(h^(-1))≤ω . We have a cover E={E_γ:γ<ω} of the B by finite or closed subspace s.t. the map h^(-1) |_(E_γ ):E_γ⟶A is continuous ∀γ<ω.

Given h is w-homeomorphism, so h is bijective. Let I be the set of cardinality |I|≤ω, we have a cover C={C_i:i∈I≤ω} for space A and a cover E={E_i:i∈I≤ω} for space B by finite or closed subsets. Therefore, the map h|_(C_i ) is homeomorphism of C_i onto E_i and the cover E has the property ρ (by 1.8). Hence, B preserved the property ρ. 

Results

Theorem 3.1.  If B is regular and h:A⟶B is s-continuous, then for each subspace X≠⌀ of A the set C(h|_X) of the map h|_X:X⟶h(X) has non-empty interior in the subspace X. 

Proof. We can assume that A=X, let M be the set of continuity points of the map h:A⟶B, Its clear that ¯M=A. Put M^*=AM and suppose that Int(M)=⌀, then ¯(M^* )=A. We fix the point of continuity x_0∈M^* of restriction map h|_(M^* ). Since x_0 is not a continuity point of the map h and B is a regular, so there is a nbd V of the point h(x_0) in B s.t for any nbd U of x_0 in A, h(U)¯V≠⌀ We fix a nbd U^* of the point x_0 in A for which h(U^*∩M)⊂V, then h(U^*∩M)¯V≠⌀ 

Put F={y∈U^*∩M:h(B)∈B¯V}, because F⊂M there is nbd P of F in A such that h(P)⊂B¯V and because the ¯(M^* )=A and P∩U^*≠⌀, then P∩U^*∩M^*=⌀. But h(P∩U^*∩M)⊂V∩(B¯V)=⌀ which is contradiction, therefore for each X≠⌀ the set C(h|_X) of the map h|_X:X⟶h(X) has a non-empty interior in the subspace X. 

Theorem 3.2  Let h:A⟶B be a s-continuous surjective map from space A onto a regular space B and ρ be a c-hereditary σ-additive projective class of topological spaces, if A∈ρ and 

¯dec(h)

Proof. Given h is surjective s-continuous function and A∈ρ. By the definition of ¯dec(h), we have a cover {C_β:β<¯dec(h)} of A by finite or closed subsets s.t. ∀β<¯dec(h) the map h|_(C_β ) is continuous. Since given A has σ-additive property, so the closed cover {C_β:β<¯dec(h)} is countable and {C_β:β<¯dec(h)}⊂ρ.

The map h|_(C_β ):C_β⟶B is surjective continuous for each β

Theorem 3.3  Let ρ be a c-hereditary ¯dec(h)-additive projective class of topological spaces. If A∈ρ and h:A⟶B is a s-continuous surjective map, then B∈ρ. 

Proof. Let us assume A has ¯dec(h)-additive property, so A has a cover C of closed subspaces s.t. |C|≤¯dec(h) and C∈ρ. By the definition of cardinal ¯dec(h) the map h|_C:C⟶B is continuous ∀C∈C. Since A has p-property and map h|_C is surjective continuous implies B∈ρ. Applying 1.5 to classes of regular spaces on s-continuous maps, we get

Corollary 3.1  If A has h-lindelof and σ-compact and h: A⟶B is a surjective s-continuous function between regular spaces, then B has that property, too. 

Theorem 3.4  If A has c-hereditary κ-summable p-property ρ and h:A⟶B is a p-wise discontinuous surjective map, then B preserved property ρ, too. 

Proof. Given h is p-wise discontinuous, so A has a countable closed cover C s.t. the map h|_C:C⟶B is continuous ∀C∈C.

Consider the topological sum of the family C as 

⊕┬  C=⋃_(C∈C)▒‍{C}×C

Since given A have c-hereditary, i.e. every finite or closed subspace of A preserved the property ρ and also given A has |C|-summable, therefore topological sum ⊕_  C preserved the property ρ. Now consider a surjective continuous map 

H:⊕┬  C⟶B

H(C,a)=h(a)    ∀C∈C    and    a∈C

Since H is surjective continuous map and ⊕_  C has p-property ρ. Hence B=h(A) has property ρ. 

Applying 2.5 to κ-additive, we get 

Corollary 3.2 If A has c-hereditary κ-additive p-property ρ and h:A⟶B is a p-wise discontinuous surjective map, then B preserved property ρ, too. 

Theorem 3.5  If A second countable and h:A⟶B is a w-discontinuous surjective function, then h(A) lindelof space. 

Proof. Let X be an arbitrary subspace of A, since A is second countable and X is subspace of A, so X is also second countable. Let U be the dense subset of subspace X in A, the closure of U is also second countable space (X=¯U). Since U⊂¯U, hence U is second countable.

Since h:A⟶B is w-discontinuous, so the restriction map h|_U:U⟶B is continuous and U is lindelof, hence h(B) is lindelof space. 

Theorem 3.6  Let ρ be κ-additive c-hereditary top-property, where 

κ=max{par(A),par(B),ϕ(A),ϕ(B)}

Let A and B are regular and s-homeomorphic perfectly para-compact spaces, if A is σ-compact with property ρ, then B has that property, too. 

Proof. Let us assume A and B are s-homeomorphic perfectly para-compact spaces and A has κ-additive property, then it has a cover C={A_i:i∈κ} of closed subspaces with property ρ s.t. |C|≤κ. Since, B is σ-compact, therefore B can be written us union of countable many compact subsets, i.e. B has a cover D={B_i:i∈κ} of closed subsets.

We find covers C and D for spaces A and B respectively, by finite or closed subsets, therefore for each i∈κ the map h|_(A_i ) is homeomorphic of A_i onto B_i (by 1.9). Hence, B has property ρ. Combining 2.9 and 1.11, we get following corollary. 

Corollary 3.3  Let h:A⟶B be a s-homeo morphism between regular spaces A and B and let ρ be κ-additive c-hereditary top-property. Then A preserved property ρ iff B has property ρ, where 

κ=hl(A).hl(B).ϕ(A).ϕ(B). 

Let X=[0;1)be the subspace of the real line with ordinary topology and Y=([0;1),τ) where τ is a topology generated by the base β={[a;b):a,b∈[0;1),a

Conclusion

This paper investigates the preservation of topological properties under weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps. It examines conditions for invariance and establishes significant relationships among these mapping classes. Additionally, it explores the implications for corresponding homeomorphisms in maintaining key topological characteristics. The findings enhance the understanding of continuity and discontinuity in topological spaces, offering insights into structural stability. The results lay a foundation for further studies on generalized continuity and its applications. Future research may extend these investigations to more complex topological structures and explore their relevance in functional analysis and related mathematical disciplines.

Authors Contribution

S.B.: Conceptualized the study, formulated the research problem, and contributed to the writing and structuring of the manuscript. M.D.: Provided supervision, reviewed the mathematical proofs for accuracy, and assisted in refining theoretical discussions. B.A.S.: conducted the literature review, and assisted in manuscript preparation. All authors collaborated on reviewing and revising the final manuscript and approved its submission. 

Acknowledgement

We sincerely express our gratitude to Assistant Professor Mustafa Danesh for his invaluable assistance in reviewing our work for grammatical accuracy and scientific rigor. His insightful feedback greatly contributed to improving the clarity and quality of our research. We also extend our appreciation to the editors and referees for their constructive comments and thoughtful suggestions, which played a crucial role in refining our study. Their expertise and meticulous review process have been instrumental in shaping the final version of this work. After an extensive review process and numerous revisions, we are delighted to see our research published online. This journey has been both challenging and rewarding, and we are grateful for the collective efforts that have made this publication possible.

Conflicts of Interest

The authors confirm that there are no conflicts of interest associated with this research. 

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Article Info:

Academic Editor

Dr. Toansakul Tony Santiboon, Professor, Curtin University of Technology, Bentley, Australia

Received

March 18, 2025

Accepted

April 21, 2025

Published

April 28, 2025

Article DOI: 10.34104/ajpab.025.01600168

Corresponding author

Suhaila Barekzai*

Department of Mathematics, Kabul Education University, Kabul City, Afghanistan

Cite this article

Barekzai S, Danesh M, and Salimi BA. (2025). Preservation of topological properties under some Maps. Aust. J. Eng. Innov. Technol., 7(2), 160-168. https://doi.org/10.34104/ajpab.025.01600168   

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