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Keywords:
monotonically star compact; regular closed; perfect; star-compact; covering; star-covering; topological space
monotonically star compact; regular closed; perfect; star-compact; covering; star-covering; topological space
Summary:
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}\colon 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 $X$. \endgraf The above statement (2) gives an answer to the question of Song (2015).
References:
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