## 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

Zbl 0931.54019