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On Epi -Regular Spaces IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 17, Issue 3 Ser. II (May – June 2021), PP 41-48 www.iosrjournals.org On Epi 휶 -Regular Spaces NADIA GHEITH1 and ALAMIN ABUSBAIHA2 1 Mathematics Department/University of Gharyan, Gharyan, Libya 2Mathematics Department/University of Gharyan, Mezda, Libya Abstract: A topological space (푋, 휏) is said to be epi 훼-regular if a coarser topology 휏′ on 푋 exists such that ′ 푋, 휏 is 푇1, 훼-regular. In this article we introduce and implement this property and give some examples to show the relationships between epi 훼 -regular, epiregular, epinormal, submetrizable semiregular and Almost 훼- normal (almost 훽-normal). Keywords: Epiregular, Epi 훼-regular, Epinormal, Semiregular, Submetrizable, and Almost 훼-normal (almost 훽-normal). Mathematics Subject Classification: 54퐷15 − 54퐵10. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 30-05-2021 Date of Acceptance: 13-06-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction A topological space (푋, 휏) is said to be epiregular [4] if a coarser topology 휏 on 푋 exists such that ′ 푋, 휏 is 푇1, regular . A topological space (푋, 휏) is said to be 훼 -regular [10], [30] if for every closed subset 퐹 of 푋 and 푥 ∈ 푋 such that 푥 ∉ 퐹 there exist disjoint open sets 푈 and 푉 such that 푥 ∈ 푈 and 푉 ∩ 퐹 is dense in 퐹. We use these definitions to introduce another new topological property as a simultaneous generalization called epi 훼 -regularity. The intent of this article is to implement this property. We show the relationship between epi 훼 - regular space, 훼 -normal, 훼 -regular, epinormal, epiregular, semiregular and Almost α-normal (almost β-normal) spaces. Also we show that every epi 훼 -regular space is Hausdorff. We prove that submetrizability or 푇1, 훼 - regularity imply epi 훼 -regularity but the converse is not correct in general. We give some examples to show that epi 훼 -regularity, 훼 -regularity and semiregularity are not necessarily related. II. Epi 휶 -Regularity Definition 1.1. A topological space (푋, 휏) is said to be epi 훼 -regular if a coarser topology 휏′ on 푋 exists such ′ that 푋, 휏 is 푇1, 훼 -regular. Note that if we necessarily let 푋, 휏′ to be just 훼 -regular in the above definition, then any space will be epi 훼 -regular since the indiscrete topology will satisfy the property. ′ Observe that if for any topological space 푋, 휏 which is 푇푖, 푖 ∈ {0,1,2} then any larger topology 휏 on 푋 so is, and since every 훼 -regular 푇1 is Hausdorff [10], [30], then we can conclude the following. Theorem 1.2. Every epi 훼 -regular space is Hausdorff.■ We note that if 푋 is not 푇푖, where 푖 ∈ {0,1,2}, then 푋 is not epi 훼 regular. For example, Sierpinski space and The closed extension topology see [9], are not Hausdorff, then they cannot be epi 훼 -regular. Since every regular space is 훼 -regular, then the next theorem is true. Theorem 1.3. Every epiregular space is epi 훼 -regular.■ The opposite direction of the above statement is not always true, but we still have the following correct. Theorem 1.4. If (푋, 휏) is an epi 훼 -regular space, and the witnesses of epi 훼 -regularity 푋, 휏′ is first countable, then (푋, 휏) is epiregular.■ Before proofing the above theorem, we need the following proposition which is proved by a similar argument found in [29]. Proposition 1.5. [30] Every first countable 훼 -regular Hausdorff space is regular. Proof. Using a contradiction, we suppose that 푋 is a first countable, Hausdorff and non regular space. Then there is an 푥 ∈ 푋 and a closed subset 퐴 of 푋 such that 푥 ∉ 퐴 where there are no disjoint open sets that separate them. Let 푈푛 : 푛 ∈ 휔 be an open base in 푥 such that 푈푛+1 ⊂ 푈푛 for all 푛 ∈ 휔. Let 퐻 = 푥푛 : 푥푛 ∈ 푈 푛 ∩ 퐴, 푛 ∈ 휔 . Note that 푥푛 was chosen inductively and because the space 푋 is Hausdorff, we can also suppose at each step of the induction that 푥푛 ∉ 푈 푛 + 1 , it follows that 푥푛 ∈ 푈 푚 if and only if 푚 ≤ 푛. DOI: 10.9790/5728-1703024148 www.iosrjournals.org 41 | Page On Epi 훼 -Regular Spaces ∘ ∘ The set 퐻 is closed. Indeed, if 푦 ∉ 퐻 , then 푋 ∖ 푈 푛 ∩ 퐴 is an open set containing 푦 and not intersecting 퐻 which implies that 푋 ∖ 푈 푛 ∩ 퐴 is a neighborhood open set containing 푦 and not intersecting 퐻. Therefore 푦 ∉ 퐻. Note that 푥 ∉ 퐻. Since 푥 and 퐻 can not be separated, so 푋 is not an 훼-regular space. ■ Proof of theorem (ퟏ. ퟒ): It is straightforward by proposition 1.5 and theorem 1.3.■ Theorem 1.6. If (푋, 휏) is an epi 훼-regular space, and the witnesses of epi 훼 -regularity 푋, 휏′ is first countable, then (푋, 휏) is completely Hausdorff. Proof. Let (푋, 휏) be any epi 훼-regular space, and let 푥, 푦 be any distinct points in 푋, then one can find a coarser ′ ′ ′ topology 휏 on 푋 such that 푋, 휏 is 푇1, 훼 -regular, and then 푋, 휏 is Hausdorff [10]. It follows that there exist two disjoint open sets 퐺, 퐻 ∈ 휏′ such that 푥 ∈ 퐺, 푦 ∈ 퐻,. Now since 푋, 휏′ is first countable then by proposition ′ ′ 1.5 푋, 휏′ is regular, so there exist 푈, 푉 ∈ 휏′ such that 푥 ∈ 푈 ⊆ 푈 휏 ⊆ 퐺 and 푦 ∈ 푉 ⊆ 푉 휏 ⊆ 퐻, where ′ ′ ′ 푈 휏 = 푥 ∈ 푋: 푊 ∩ 푈 ≠ ∅, ∀ open 푊 in 휏 ′ , 푥 ∈ 푊 similarly 푉 휏 . Since 퐴 휏 ⊆ 퐴 휏 , for any 퐴 ⊆ 푋, this implies 휏 휏′ 휏′ 휏′ That 푈 ⊆ 푈 . 퐴s 푈 ∩ 푉 = 0. Thus (푋, 휏) is completely Hansdorff. ■ Thus any space (푋, 휏) which is not completely Hausdorff, such that any coarser topology of it is 푇2 first countable, cannot be epi 훼 -regular. Since any 훽 -normal or 훼 -normal [26] satisfying 푇1 axiom is 훼-regular [30], [10], then we end to the following theorem Theorem 1.7. Every epi 훽 -normal (epi 훼 -normal) space is epi 훼 -regular. ■ As every second countable 푇3 space is metrizable, [[8],4.2.9], and since every second countable is first countable then by proposition 1.5 we have the following corollary. Corollary 1.8. If (푋, 휏) is epi 훼 -regular and the witness of epi 훼-regularity 푋, 휏′ is second countable, then (푋, 휏) is submetrizable. ■ Note that corollary 1.8 is not correct in general. For example, the Tychonoft Plank 휔1 + 1 × 휔0 + 1 ∖ 휔1, 휔0 is Tychonoff being Hausdorff locally compact, and hence it is epi 훼 -regular, but it is not submetrizable, because if it was, then 휔1 + 1 × {0} ⊆ 휔1 + 1 × 휔0 + 1 ∖ 휔1, 휔0 is submetrizable, because submetrizablity is hereditary, but 휔1 + 1 × {0} ≅ 휔1 + 1 and 휔1 + 1 is not submetrizable. It is well known that 푇2 paracompact space is 푇4, then we have the following result proved. Corollary 1.9. If (푋, 휏) is epi 훼-regular and the witness of epi 훼 -regularity 푋, 휏′ is paracompact, then 푋, 휏 is 푇4. ■ Also, we remind that any 푇2 compact space is 푇4, and we conclude. Corollary 1.10. Any epi 훼-regular compact space is 푇4. ■ A Hausdorff space 푋 is said to be 퐻-closed if 푋 is a closed subspace of every Hausdorff space in which it is contained [[8],3.12.5]. Since a regular space is 퐻 -closed if and only if it is compact [[8],3.12.5]. Then we can prove a similar argument for epi 훼 -regularity. Corollary 1.11. If (푋, 휏) is epi 훼 -regular compact space, then the witness of epi 훼 -regularity 푋, 휏′ is 퐻- closed. ■ Theorem 1.12. If (푋, 휏) is an epi 훼 -regular space, then for every compact subset 퐹 of 푋 and every 푥 ∈ 푋 such that 푥 ∉ 퐹, there exist disjoint open sets 푈, 푊 such that 퐹 ∩ 푈 = 퐹 and 푥 ∈ 푊. Proof. Let (푋, 휏) be an epi 훼 -regular space, then a coarser topology 휏′ on 푋 exists such that 푋, 휏′ is 훼 - ′ regular, 푇1. Let 퐹 be any compact set in (푋, 휏) and let 푥 ∉ 퐹, hence 퐹 is closed in 푋, 휏 and 푥 ∉ 퐹, by 훼- regularity of 푋, 휏′ , there exist 푈, 푊 ∈ 휏′ such that 퐹 ∩ 푈 = 퐹, 푥 ∈ 푊 and 푈 ∩ 푊 = ∅. ■ Corollary 1.13. If 퐹 and 퐸 disjoint compact sets in an epi 훼-regular space 푋, then there exist disjoint open sets 푈 and 푊 such that 퐹 ∩ 푈 = 퐹, 퐸 ∩ 푊 = 퐸. Proof. Let (푋, 휏) be an epi 훼 -regular space, then there exists a coarser topology 휏′ on 푋 such that 푋, 휏′ is 훼 - ′ regular, 푇1. Let 퐹, 퐸 be any disjoint compact subsets of (푋, 휏), hence they are disjoint compact subsets of 푋, 휏 and by theorem 1.12 for each 푎 ∈ 퐹 and compact set 퐸, there exist open sets 푈푎 , 푊푎 such that 푎 ∈ 푈푎 , 퐸 ∩ 푊 푎 = 퐸 and 푈푎 ∩ 푊푎 = ∅. Now consider 퐹 is an arbitrary compact set disjoint from 퐸. For each 푎 in 퐹, by theorem 1.12 gives disjoint open sets 푈푎푖 containing 푎 and 퐸 ∩ 푊푎푖 = 퐸 and 푈푎푖 ∩ 푊푎푖 = 0. The family 푈푎푖 : 푖 ∈ 퐼 is an open cover of 퐹, since 퐹 is compact, there is a finite subfamily 푈푎1 , … , 푈푎푛 which covers 퐹 and the 푛 corresponding 푊푎1 , … , 푊푎푛 is a closed cover of 퐸.
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