Hα-Compact and Hα-Lindelof Topological Spaces

Abstract

Most references on topological spaces seem to define a compact space in terms of open coverings of a space having finite subcoverings. Formally, we say that a collection of open sets  {Oi:i∈⋀} indexed by some set ⋀ is an open covering of a topological space (Z, T) if  Z⊆∪{Oi:i∈⋀}.  We define Z to be compact if there exist finitely many open sets O1, O2, O3, . . . , On in the open covering such that  Z⊆∪{Oi:i=1, 2, 3, . . . , n∈⋀}. There exist several equivalent definitions of a compact space, which we briefly mention here. A topological space Z is compact if and only if Z has the finite intersection property: if {Oi:i∈⋀} is a collection of a closed set indexed by a set ⋀ such that, for any finite subset ∆⊆⋀, ⋂{Oi:i∈∆}≠∅, then ⋂{Oi:i∈⋀}≠∅. Another equivalent definition is in terms of nets. A topological space Z is compact if and only if every net {zi:i∈⋀}  in Z has a convergent subnet.  In 2022, Beyda S. Abdullah, Sabih W. Askandar and Ruqayah N. Balo introduced a new kind of open set, which is called an hα-open  set. They also studied the relationship between hα-open sets and a variety of other open sets as h-open sets, open sets, semi-open sets, and α-open sets. Furthermore, they introduced the concepts of hα-continuous mappings, hα-open mappings, hα-irresolute mappings, and hα-totally continuous mappings. We will extend the concept of compactness via hα-open sets by introducing hα-compact spaces in topological spaces and will investigate their characterizations by making use of generalized mappings, including hα-continuous and  hα-irresolute functions. The objective of this paper is to introduce the new concepts called hα-compact space, hα-Lindelof space, countably hα-compact space, almost hα-compact space, and mildly hα-compact space in topological spaces and investigate fundamental properties and characterizations of these new notions of spaces.