Abstract
We give necessary and sufficient conditions for a compact (countably compact) set to be closed in S 2 (Fréchet, S 2) and in normal (Fréchet, normal) spaces. Sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets, in such spaces. Countable compactness of the closure of a countably compact set in Fréchet, S 2-spaces, and related results are also obtained.
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The research of the second author is supported hy C.S.I.R. Award No. 2-3I/97(i)-E.U.II, New Delhi, India.
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Garg, G.L., Singh, N. When a compact (countably compact) set is closed. Acta Math Hung 94, 233–239 (2002). https://doi.org/10.1007/s10474-002-0006-3
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DOI: https://doi.org/10.1007/s10474-002-0006-3
Key words and phrases
- compact
- countably compact
- closed
- S 2
- Fréchet
- normal
- closure
- T 2
- second countable
- T 0
- T 1
- S 1
- regular
- sequence
- neighborhood
- net
- cluster point