Pseudo-Normality and Pseudo-Tychonoffness of Topological Groups - Mesfer H. Alqahtani; Hanan Al-Saadi; Eman Alluqmani; Zanyar A. Ameen (Mathematics)
It is common knowledge that any topological group that satisfies the lowest separation axiom, <semantics>T0</semantics>, 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.