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A \(\beta \)-normal Tychonoff space which is not normal. (English) Zbl 1090.54016

The present paper answers 3 questions about \(\alpha \)-normal and \(\beta \)-normal spaces that were defined and investigated by A. V. Arhangel’skii and L. Ludwig [Commentat. Math. Univ. Carol. 42, 507–519 (2001; Zbl 1053.54030)]: 1. There exists an \(\alpha \)-normal non-regular Hausdorff space; 2. There exists a \(\beta \)-normal regular non-Tikhonov space; 3. There exists a \(\beta \)-normal non-normal Tikhonov space. It is also proved that every 1st countable \(\alpha \)-normal Hausdorff space is regular.

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Citations:

Zbl 1053.54030
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