Pseudo-Normality and Pseudo-Tychonoffness of Topological Groups

Pseudo-Normality and Pseudo-Tychonoffness of Topological Groups

It is common knowledge that any topological group that satisfies the lowest separation axiom, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>0</mn></msub></s...

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Bibliographic Details
Main Authors:	Mesfer H. Alqahtani, Hanan Al-Saadi, Eman Alluqmani, Zanyar A. Ameen
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Mathematics
Subjects:	topological group pseudo-normal topological group pseudo-Tychonoff topological group
Online Access:	https://www.mdpi.com/2227-7390/13/1/30
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Summary:	It is common knowledge that any topological group that satisfies the lowest separation axiom, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>0</mn></msub></semantics></math></inline-formula>, is immediately Hausdorff and completely regular; however, this is not the case for normality. This motivates us to introduce the concept of pseudo-normal groups along with pseudo-Tychonoff topological groups as generalizations of the normality and Tychonoffness of topological groups, respectively. We show that every pseudo-normal (resp. pseudo-Tychonoff) topological group is normal (resp. Tychonoff). Generally, the reverse implication of the latter does not hold. Then, we discuss their main properties in detail. To clarify these properties, we provide some examples. Finally, we establish some other results.
ISSN:	2227-7390