Pseudocompact $$\varDelta $$-spaces are often scattered
| Autor: | V. V. Tkachuk, Arkady Leiderman |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Monatshefte für Mathematik. 197:493-503 |
| ISSN: | 1436-5081 0026-9255 |
| DOI: | 10.1007/s00605-021-01628-3 |
| Popis: | Given a pseudocompact $$\varDelta $$ -space X, we establish that countable subsets of X must be scattered. This implies that pseudocompact $$\varDelta $$ -spaces of countable tightness are scattered. If a pseudocompact $$\varDelta $$ -space has the Souslin property, then it is separable and has a dense set of isolated points. It is shown that adding a countable subspace to a pseudocompact $$\varDelta $$ -space can destroy the $$\varDelta $$ -property. However, if X is countably compact and $$Y\subset X$$ is a $$\varDelta $$ -space for some $$Y\subset X$$ such that $$|X\backslash Y|\le \omega $$ , then X is a $$\varDelta $$ -space. We also show that monotonically normal $$\varDelta $$ -spaces must be hereditarily paracompact. Besides, if X is a subspace of an ordinal with its order topology, then X is hereditarily paracompact if and only if it has the $$\varDelta $$ -property. |
| Databáze: | OpenAIRE |
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