Between compactness and quasicompactness

Abstract

Three weak variants of compactness which lie strictly between compactness and quasicompactness, are introduced. Their basic properties are studied. The interplay with mapping and their direct and inverse preservation under mappings are investigated. In the process three decompositions of compactness are observed.

On some variants of compactness

“…In each case, we observe that there is an appropriate change of topology which reveals that the new concept is equivalent to the classical notion of compactness. The last two results of Kohli and Singh [10,Theorems 5.17 and 5.18] make this observation almost as a post script. They do not use it anywhere in their paper [10].…”

“…In 2005 Kohli and Singh [10] introduced three weak variants of compactness which lie between compactness and quasicompactness. They studied the basic properties of the classes of spaces defined by these three weak variants of compactness, which have been named d-compactness, d * -compactness and D δ -compactness.…”

Some properties of the quasicompact-open topology on C(X)

“…Let X be the skyline space [10]. The space X is a quasicompact, but not compact [14]. Hence, we obtain C k (X) ≤ C q (X) = C u (X).…”

“…Let X = N and define a topology on X by taking every odd integer to be open and a set U is open if for every even integer p ∈ U , the predecessor and the successor of p are also in U [14]. From this it follows that C k (X) ≤ C q (X) = C u (X).…”

Pseudo perfectly continuous functions and closedness/compactness of their function spaces

“…Definition 5.6. A space X is said to be D δ -compact [23] (mildly compact [57]) if every cover of X by regular F σ -sets (clopen sets) has a finite subcover.…”