Mathematica Bohemica, Vol. 147, No. 3, pp. 319-323, 2022
Remarks on monotonically star compact spaces
Sumit Singh
Received September 28, 2020. Published online July 29, 2021.
Abstract: A space $ X $ is said to be monotonically star compact if one assigns to each open cover $ \mathcal{U} $ a subspace $ s(\mathcal{U}) \subseteq X $, called a kernel, such that $ s(\mathcal{U}) $ is a compact subset of $ X $ and $ {\rm St}(s(\mathcal{U}),\mathcal{U})=X $, and if $ \mathcal{V} $ refines $ \mathcal{U} $ then $ s(\mathcal{U}) \subseteq s(\mathcal{V}) $, where $ {\rm St}(s(\mathcal{U}),\mathcal{U})= \bigcup\{U \in\nobreak\mathcal{U} U \cap s(\mathcal{U}) \not= \emptyset\} $. We prove the following statements: \item{(1)} The inverse image of a monotonically star compact space under the open perfect map is monotonically star compact. \item{(2)} The product of a monotonically star compact space and a compact space is monotonically star compact. \item{(3)} If $ X $ is monotonically star compact space with $ e(X) < \omega$, then $ A(X) $ is monotonically star compact, where $ A(X) $ is the Alexandorff duplicate of space $X$. The above statement (2) gives an answer to the question of Song (2015).
Keywords: monotonically star compact; regular closed; perfect; star-compact; covering; star-covering; topological space
