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Some topological properties on C- -Normality and C- -Normality
⁎Address: Department of Mathematics, College of science, Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia. mam_1420@hotmail.com (Samirah Alzahrani) Samar.alz@tu.edu.sa (Samirah Alzahrani)
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Received: ,
Accepted: ,
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.
Peer review under responsibility of King Saud University.
Abstract
A topological space is called C- -normal (C- -normal) if there exist a bijective function from onto -normal ( -normal) space such that the restriction map from onto is a homeomorphism for any compact subspace of . We discuss some relationships between C- -normal (C- -normal) and other properties.
Keywords
Normal
α-normal
β-normal
C-normal
Epinormal
Mildly normal
54D15
54B10
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1 Introduction
In 2017 we discuss the topological property “ -normal” (AlZahrani and Kalantan, 2017). In this paper we introduce a new property called C- -Normality and C- -Normality. We show any -normal ( -normal) space is C- -normal (C- -normal), but the converse is not true in general. And we show that any C-normal, lower compact, epinormal, epi- -normal and epi- -normal spaces is C- -normal (C- -normal). We prove any locally compact is C- -normal (C- -normal) but the converse is not true in general. Also observe that a witness function of C- -normal (C- -normal) not necessarily to be continuous in general, but it will be continuous under some conditions.
2 C– -Normality and C- -Normality
Recall that a topological space is called an -normal space (Arhangel’skii and Ludwig, 2001) if for every-two disjoint closed subsets and of there are two open subsets and of such that is dense in ; is dense in , and , and a topological space is called a -normal space (Arhangel’skii and Ludwig, 2001) if for every-two disjoint closed subsets and of there are two open subsets and of such that is dense in ; is dense in , and . A topological space is called C-normal (AlZahrani and Kalantan, 2017) if there exist a bijective function from onto a normal space such that the restriction map from onto is a homeomorphism for any compact subspace of .
A topological space is called C- -normal (C- -normal) if there exist a bijective function from onto -normal ( -normal) space such that the restriction map from onto is a homeomorphism for any compact subspace of .
In these definition, we call the space a witness of C- -normal (C- -normal) and the function is called a witness function.
A topological space is called -regular (Alzahrani, 2022) if for any and a closed subset such that there are two disjoint open sets ; such that and . And topological space is called almost -regular (Alzahrani, 2022) if for any and a regular closed subset such that there are two disjoint open sets ; such that and .
Any regular space is -regular.
Let be a regular space. Pick and be a closed set such that , then there exist two disjoint open sets and subsets of where and , hence (note that since is closed), and , therefore is -regular space.
(Alzahrani, 2022) Any -regular space is almost -regular.
From Lemma 1.2. and Lemma 1.3. we conclude the following corollary.
Any regular space is almost -regular.
Any normal space is -normal.
Lemma 1.6 (Arhangel’skii and Ludwig, 2001)
Any normal space is -normal.
Let be a normal space. Pick two disjoint closed sets and subsets of . Since is normal, then there exist two disjoint open sets and subsets of where , and . Hence and . It remains to prove . For a normal space Y, if F is a closed set, U is an open set and F ⊆ U, then there exist an open set V such that F ⊆ V ⊆ ⊆ U. Now apply this to and set = V and = Y\ .
So we have the following theorem.
Any C-normal space is C- -normal (C- -normal).
The converse is true under some conditions, first we mention some definition.
A Hausdorff space is extremally disconnected (Engelking, 1977) if the closure of any open set in is open. A topological space is called mildly normal (Shchepin, 1972) if any two disjoint regular closed subsets can be separated.
Theorem 1.8 (Arhangel’skii and Ludwig, 2001)
Any -normal extremally disconnected space is normal.
Let be a -normal extremally disconnected space. Pick two disjoint closed sets and subsets of . Since is -normal, then there exist two disjoint open sets and subsets of where and . Hence and . However, ∩ = ∅ since is open and ∩ = ∅. Thus, ∩ = ∅ as well since is open (by extremally disconnectedness) and ∩ = ∅.
Therefore is normal space.
From Theorem 1.8, we have the following.
If is - -normal ( - -normal) such that the witness of - -normal ( - -normal) is extremally disconnected, then is -normal.
If is - -normal such that the witness of - -normal is mildly normal, then is -normal.
Let be C- -normal. Then the codomain witness of C- -normal is -normal. Let and be any disjoint closed subsets of . Since is -normal, there exist open subsets and of where , and . So , are disjoint regular closed subsets containing and respectively. Since is mildly normal, there exist disjoint open subsets and of where and . Hence is normal.
Any -normal space satisfying axiom is Hausdorff.
Let be any -normal -space. Let be any two distinct elements in . Hence and are disjoint closed subsets of , by -normality, there exist two disjoint open subsets and of where and which implies and . Therefore is Hausdorff.
Lemma 1.12 (Arhangel’skii and Ludwig, 2001)
Any -normal space satisfying axiom is regular (hence Hausdorff).
By Corollary 1.4. we have the following result.
Any -normal space satisfying axiom is almost -regular.
Also by Lemma 1.12 and Lemma 1.2 we have the following result.
Any -normal space satisfying axiom is -regular.
Any -normal space satisfying axiom is -regular.
By Lemma 1.3. we conclude the following corollary.
Any -normal space satisfying axiom is almost -regular.
(Murtinov́a, 2002) Every first countable -normal Hausdorff space is regular.
Recall that a topological space (Y, τ) is called submetrizable (AlZahrani and Kalantan, 2017) if there exists a metric d on Y such that the topology τ d on Y generated by d is coarser than τ.
Every submetrizable space is C- -normal (C- -normal).
Let be a submetrizable space, the there exists a metrizable such that . Hence is -normal since it is normal, and the identity function from onto is a one-to-one and continuous function. If we take any compact subspace of , then is hausdorff, since it is subspace of , and by (Engelking, 1977);3.1.13]; is a homeomorphism.
The Rational Sequence Topology (Steen and Seebach, 1995) is submetrizable being finer than the usual topology , so is C- -normal (C- -normal).
The converse of Theorem 1.18. is not true in general, for example is C- -normal (C- -normal) which is not submetrizable.
Apparently, any -normal ( -normal) space is C- -normal (C- -normal), to prove this, just by considering and is the identity function.
While in general the converse is not true. We provide some examples below.
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The Half-Disc topological space (Steen and Seebach, 1995) is C- -normal (C- -normal) because it is submetrizable by Theorem 1.18. but it is not -normal nor -normal because it is first countable and Hausdorff but not regular, so by Proposition 1.17. the Half-Disc topological space is not -normal space, hence not -normal. In general C- -normality (C- -normality) do not imply -normality ( -normality) even with Hausdorff or first countable properties.
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The Deleted Tychonoff Plank (Steen and Seebach, 1995), it is C- -normal (C- -normal) since it is locally compact by Theorem 2.7. but it is not -normal nor -normal see (Arhangel’skii and Ludwig, 2001).
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The Dieudonn Plank (AlZahrani and Kalantan, 2017), in example 1.10 we proved that it is C-normal, hence it is C- -normal (C- -normal) by Theorem 1.9. but it is not -normal nor -normal see (Arhangel’skii and Ludwig, 2001), also not locally compact, hence this example also shows that the converse of Theorem 2.7. is not true.
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The Sorgenfrey line square see (Steen and Seebach, 1995) is not normal, but it is submetrizable space being it is finer than the usual topology on , so by Theorem 1.18. it is C- -normal (C- -normal).
If is a compact non– -normal (non– -normal) space, then can not be C- -normal (C- -normal).
Assume is a compact non– -normal (non– -normal) space. Suppose is C- -normal (C- -normal), then there exists -normal ( -normal) space and a bijective function where the restriction map from onto is a homeomorphism for any compact subspace of . As is compact, then , and we have a contradiction as is -normal ( -normal) while is not. Hence can not be C- -normal (C- -normal).
Observe that a function witnessing of C- -normal ( - -normal) of not necessarily to be continuous in general, and here is an example.
Let with the countable complement topology (Steen and Seebach, 1995). We know is and the only compact sets are finite, hence the compact subspaces are discrete. If we let be the discrete topology on , then obviously the identity function from onto is a witnessing of the C- -normality (C- -normality) which is not continuous.
But it will be continuous under some conditions as the following theorems.
If is a C- -normal (C- -normal) and Fréchet space, then any function witnessing of C- -normality (C- -normality) is continuous.
Let be a Fréchet C- -normal (C- -normal) space and be a witness of the C- -normality (C- -normality) of . Let and pick . There is a unique where thus . since is Fréchet, then there exists a sequence where . As the subspace of is compact, the induced map is a homeomorphism. Let be any open neighborhood of . Then is an open neighborhood of in the subspace . Since is a homeomorphism, then is an open neighborhood of y in K, then there exists where hence then Hence and Thus is continuous.
Since any first countable space is Fréchet, we conclude that, In C- -normality (C- -normality) first countable space a function is a witness of the C- -normality (C- -normality) of is continuous. Also, by theorem (Engelking, 1977),3.3.21], we conclude the following.
If is a C- -normal (C- -normal) -space and is a witness function of the C- -normality (C- -normality), then is continuous.
For simplicity, let us call a space which satisfies that the only compact subspaces are the finite subsets F-compact. Clearly F-compactness is a topological property.
If is F-compact, then is - -normal ( - -normal).
Let be a F-compact. Let and let with the discrete topology. Hence the identity function from onto does the job.
Consider , where is the countable complement topology (Steen and Seebach, 1995). We know is and the only compact sets are finite, therefore, by Theorem 1.25. is - -normal ( - -normal). This a fourth example of - -normal ( - -normal) but not -normal (nor -normal).
Notice that any topology finer than a topological space is . Also any compact sub set of a topological space is compact in any topology coarser than on .
Hence any topology finer than F-compact topological space is also F-compact. As an example, denotes the Fortissimo topology on , see [14, Example 25]. We know that is finer than which is F-compact, hence F-compact too. Thus, is - -normal ( - -normal).
C- -normality (C- -normality) is a topological property.
Let be a C- -normal (C- -normal) space and let . Let be a -normal ( -normal) space and let be a bijective function where the restriction map from onto is a homeomorphism for any compact subspace . Let be a homeomorphism. Hence and satisfy the requirements.
3 C– -Normality (C- -Normality) and some other properties
A topological space is called C- -regular if there exists a bijective function from onto -regular space such that the restriction map from onto is a homeomorphism for any compact subspace of .
This definition is new and we will study some of its properties later.
If is C- -normal (C- -normal) space and the witness of the C- -normality (C- -normality) of is , then is C- -regular.
We prove this corollary by Lemma 1.11, Lemma 1.12. We defined C-regular in (AlZahrani, 2018).
If is C- -normal space and the codomain witness of the C- -normality of is , then is C-regular.
We prove this corollary by Lemma 1.12.
If is a C- -normal (C- -normal) Fréchet space and the witness of the C- -normality (C- -normality) is , then is .
Let is a C- -normal (C- -normal) Fréchet space, then there exist -normal ( -normal) space (witness of the C- -normality (C- -normality)) and a bijective function such that the restriction map from onto is a homeomorphism for any compact subspace of , then by Theorem 1.23. is continuous. Let any be such that , then , . Since is -normal ( -normal) and , then by Lemma 1.11 (Lemma 1.12) the space is , then there exist and are open sets in where and . Since are open sets in and is continuous, then and are open sets in , and . Hence is .
Any C-regular Fréchet Lindelof space is C- -normal (C- -normal).
Let be any C-regular Fréchet Lindelof space. Let be a regular space and be a continuous bijective function see Theorem 1.23. By (Engelking, 1977); 3.8.7] is Lindelof. Since any regular Lindelof space is normal (Engelking, 1977), 3.8.2]. Hence is C- -normal (C- -normal).
C- -normality (C- -normality) does not imply C- -regularity nor C-regular, for example.
Consider the real numbers set with its right ray topology , where . As any two non-empty closed sets must be intersect in , then it is normal, and by Lemma in above, it is -normal ( -normal), hence C- -normal (C- -normal). Now, suppose that is C- -regular. Take -regular space and a bijective function from onto where the restriction map from onto is a homeomorphism for any compact subspace of . We know that a subspace of is compact if and only if has a minimal element. Hence is compact, then is a homeomorphism, it means as a subspace of is -regular which is a contradiction, since is closed in subspace and , but any non-empty open sets on must intersect. Then cannot be C- -regular (C-regular).
Recall that a topological space is called Locally Compact (AlZahrani and Kalantan, 2017) if is Hausdorff and for every and every open neighborhood of there exists an open neighborhood of such that and is compact.
Every locally compact space is C- -normal (C- -normal).
Let be locally compact space. By (Engelking, 1977), 3.3.D], there exists compact space and hence -normal ( -normal), and a continuous bijective function . We have from onto is a homeomorphism for any compact subspace of , because continuity ,1–1 and onto are inherited by g, also is closed since is compact and g(K) is .
Consider , the first uncountable ordinal, we consider as an open subspace of its successor , which is compact and hence is locally compact [14, Example 43]. Thus, is locally compact as an open subspace of a locally compact space, see (Engelking, 1977),3.3.8]. Then by Theorem 2.7. is C- -normal (C- -normal).
The converse of Theorem 2.7. is not true in general. We introduce the following example of C- -normal (C- -normal) which is not locally compact.
Consider the quotient space . Let , where . Define as follows:
Now consider with the usual topology . Define the topology on . Then is a closed quotient mapping. We explain the open neighborhoods of any element in as follows: The open neighborhoods of each are where is a natural number. The open neighborhoods of are , where is an open set in such that . It is clear that is , but it is not locally compact . is a continuous image of with its usual topology, so it is Lindelof and , then is . Hence it is C- -normal (C- -normal).
A topological space is called Epi- -normal (Gheith and AlZahrani, 2021) if there is a coarse topology on such that is -normal and . A topological space is called Epi- -normal (Gheith and AlZahrani, 2021) if there is a coarse topology on such that is -normal and . We defined Epinormal in (AlZahrani and Kalantan, 2016). By the same argument of Theorem 1.18. we can prove the following corollary.
Every epinormal space is C- -normal (C- -normal).
Every epi- -normal (epi- -normal)space is C- -normal(C- -normal).
Any indiscrete space which has more than one point is an example of a C- -normal (C- -normal) space which is not epi- -normal (epi- -normal).
The converse of Corollary 2.9 is true with Fréchet property.
Any C- -normal (C- -normal) Fréchet space is epi- -normal (epi- -normal).
Let be any C- -normal (C- -normal) Fréchet space. Let be -normal ( -normal) and be a bijective function. Since is Fréchet, is continuous (see Theorem 1.23). Define . Obviously, is a topology on coarser than such that is continuous. Also is open, since if we take , then where . Thus which gives that is open. Therefore is a homeomorphism. Thus is -normal ( -normal). Hence is epi- -normal (epi- -normal).
A topological space is called lower compact (Kalantan et al., 2019) if there exists a coarser topology on such that is -compact.
Any lower compact space is C- -normal (C- -normal).
Let is lower compact, then is -compact, hence normal and the identity function is a continuous and bijective. If we take any compact subspace of , then is a homeomorphism by (Engelking, 1977);3.1.13].
In general, the converse of Theorem 2.13. is not true, for example consider a countable complement topology on an uncountable set, it is C- -normal (C- -normal) since it is F-compact, but it is not lower compact because it is not .
If is C- -normal compact Fréchet space and the witness of the C- -normality is , then is lower compact.
Pick -normal space and a bijective function such that is a homeomorphism for any compact subspace . Since is Fréchet, then is continuous. Hence is compact. Since is -normal space, then by Lemma 1.11. it is Hausdorff. Hence is compact. Define a topology on as follows Then is coarser than and is a bijection continuous function. Let any , then is of the form for some . Hence . Thus is open. Hence is a homeomorphism. So is compact. Therefore is lower compact.
If is C- -normal compact Fréchet space and the witness of the C- -normality is , then is lower compact.
4 Conclusion
The aim of this paper is to introduce a new weaker version of normality called C- -normal and C- -normal. We show that some relationships between this a new topological property and some other topological properties, and there are still many topological properties that the researcher can study in this topic.
Acknowledgments
This research received funding from Taif University Researchers Supporting Project number (TURSP-2020/207), Taif University, Taif, Saudi Arabia.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References
- Arhangel’skii, A., Ludwig L. D., 2001. On α-Normal and β-Normal Spaces, Comment. Math. Univ. Carolinae. 42(3),507-519.
- General Topology. Warszawa: PWN; 1977.
- A β-Normal Tychonoff Space Which is Not Normal. Comment. Math. Univ. Carolinae.. 2002;43(1):159-164.
- [Google Scholar]
- Real functions and spaces that are nearly normal, siberian Math. J.. 1972;13:820-829.
- [Google Scholar]
- Countrexample in Topology. INC, New York: Dover Publications; 1995.
Appendix A
Supplementary data
Supplementary data to this article can be found online at https://doi.org/10.1016/j.jksus.2022.102449.
Appendix A
Supplementary data
The following are the Supplementary data to this article: