Abstract
A topological space(X,τ)is said to be strongly compact if every preopen cover of(X,τ)admits a finite subcover. In this paper, we introduce a new class of sets called -preopen sets which is weaker than both open sets and -open sets. Where a subsetAis said to be -preopen if for eachx∈Athere exists a preopen setUxcontainingxsuch thatUx−Ais a finite set. We investigate some properties of the sets. Moreover, we obtain new characterizations and preserving theorems of strongly compact spaces.
Highlights
It is well known that the effects of the investigation of properties of closed bounded intervals of real numbers, spaces of continuous functions, and solutions to differential equations are the possible motivations for the formation of the notion of compactness
Many researchers have pithily studied the fundamental properties of compactness and the results can be found in any undergraduate textbook on analysis and general topology
A topological space X is said to be strongly compact if every preopen cover of X admits a finite subcover
Summary
It is well known that the effects of the investigation of properties of closed bounded intervals of real numbers, spaces of continuous functions, and solutions to differential equations are the possible motivations for the formation of the notion of compactness. A topological space X is said to be strongly compact if every preopen cover of X admits a finite subcover. The notion of strongly compact relative to a topological space X was introduced by Mashhour et al. Quite recently Jafari and Noiri 5, 6 , by introducing the class of firmly precontinuous functions, found some new characterizations of strongly compact spaces. They obtained properties of strongly compact spaces by using nets, filterbases, precomplete accumulation points. The notion of preopen sets plays an important role in the study of strongly compact spaces. By using N-preopen sets, we obtain new characterizations and further preservation theorems of strongly compact spaces. The complement of an N-open subset is said to be N-closed
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