## 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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