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A note on k-c-semistratifiable spaces and strong \(\beta \)-spaces. (English) Zbl 1249.54063

Recall that a space \(X\) is a c-semistratifiable (CSS) space, if the compact sets of \(X\) are \(G_{\delta }\)-sets in a uniform way. In this note, the authors introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. They discuss some properties of k-c-semistratifiable spaces and prove that a \(T_2\)-space \(X\) is a k-c-semistratifiable space if and only if \(X\) has a g-function satisfying certain conditions. Using this result, the authors show that if \(X\) is a submesocompact locally k-csemistratifiable space, then \(X\) is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. In the last part of the paper, the following result is shown: if \(X\) is the countable union of closed strong \(\beta \)-spaces \(X_n\) then \(X\) is a strong \(\beta \)-space.

MSC:

54E20 Stratifiable spaces, cosmic spaces, etc.
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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