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On some questions on star covering properties. (English) Zbl 0954.54009

Summary: We prove the following statements: (1) There exist two countably compact Tikhonov spaces \(X\) and \(Y\) such that \(X\times Y\) is \(1{1\over 2}\)-starcompact but not countably compact. (2) If \({\mathfrak d}={\mathfrak b}\), then there exists a \(1{1 \over 2}\)-starcompact Tikhonov space having a zero-set which is not pseudocompact. (3) There exists a \(1{1 \over 2}\)-starcompact Hausdorff space having a zero-set which is not pseudocompact. (4) There exists a Tikhonov star-Lindelöf space \(X\) having the property (a) such that the extent \(e(X)={\mathfrak c}\). The statement (1) gives an answer to M. V. Matveev [ A survey on star-covering properties, Topological Altas, Question 26], (2) and (3) answer negatively M. V. Matveev [A survey on star-covering properties, Topological Altas, Question 20] and (4) answers negatively M. Bonanzinga [Quest. Answers Gen. Topology 16, No. 2, 79-104 (1998; Zbl 0931.54019), Question 4.5].

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10 Product spaces in general topology
54D55 Sequential spaces

Citations:

Zbl 0931.54019
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