Paranormality in topological products. (English. Russian original) Zbl 1383.54015
Math. Notes 102, No. 3, 431-433 (2017); translation from Mat. Zametki 102, No. 3, 477-480 (2017).
From the text: We consider the following weak form of the normality topological property: a space is said to be paranormal (in the sense of P. Nyikos [Topol. Proc. 9, No. 2, 367–373 (1984; Zbl 0576.54001)]) if, for any countable discrete system \(\{D_n:n<\omega\}\) of closed sets, there exists a locally finite system \(\{U_n:n<\omega\}\) of open sets such that \(D_n\subset U_n\) for all \(n<\omega\) and \(D_m\cap U_n\neq\varnothing\) if and ony if \(D_m= D_n\). It is easy to see that not only all normal spaces but also all countably paracompact spaces are paranormal.
Definition. A space \(X\) is said to be \(F_\sigma\)-paranormal if any \(F_\sigma\)-set in \(X\) is paranormal.
Zenor proved that if all \(F_\sigma\)-sets in a product \(X\times Y\) are countably paracompact, then either \(X\) is normal or all countable subsets of \(Y\) are closed. The following theorem strengthens this result of Zenor.
Theorem 1. If a product \(X\times Y\) is \(F_\sigma\)-paranormal, then either \(X\) is normal and countably paracompact or all countable subsets of \(Y\) are closed.
The main result of this note is Theorem 2, which generalizes well-known theorems by M. Katěov [Fundam. Math. 35, 271–274 (1948; Zbl 0031.28301)] and P. Zenor [Proc. Am. Math. Soc. 30, 199–201 (1971; Zbl 0222.54025)].
Theorem 2. If a product \(X\times Y\) is hereditarily paranormal, then either \(X\) is perfectly normal or all countable subsets of \(Y\) are closed.
Theorem 2 has the following immediate corollary.
Theorem 3. Any compact space with hereditarily paranormal cube is metricable.
N. Noble [Trans. Am. Math. Soc. 160, 169–183 (1971; Zbl 0233.54004)] proved that if any power of a space \(X\) is normal, then \(X\) is compact. To be more precise, if \(X^\tau\) is normal for some uncountable \(\tau\geq w(X)\), then \(X\) is compact. Here \(w(X)\) denotes the weight of the space \(X\). As an application of Theorem 1, we obtain the following generalization of Noble’s theorem [loc. cit.].
Theorem 4. If \(X^\tau\) is an \(F_\sigma\)-paranormal space for some uncountable \(\tau\geq w(X)\), then the space \(X\) is compact.
Definition. A space \(X\) is said to be \(F_\sigma\)-paranormal if any \(F_\sigma\)-set in \(X\) is paranormal.
Zenor proved that if all \(F_\sigma\)-sets in a product \(X\times Y\) are countably paracompact, then either \(X\) is normal or all countable subsets of \(Y\) are closed. The following theorem strengthens this result of Zenor.
Theorem 1. If a product \(X\times Y\) is \(F_\sigma\)-paranormal, then either \(X\) is normal and countably paracompact or all countable subsets of \(Y\) are closed.
The main result of this note is Theorem 2, which generalizes well-known theorems by M. Katěov [Fundam. Math. 35, 271–274 (1948; Zbl 0031.28301)] and P. Zenor [Proc. Am. Math. Soc. 30, 199–201 (1971; Zbl 0222.54025)].
Theorem 2. If a product \(X\times Y\) is hereditarily paranormal, then either \(X\) is perfectly normal or all countable subsets of \(Y\) are closed.
Theorem 2 has the following immediate corollary.
Theorem 3. Any compact space with hereditarily paranormal cube is metricable.
N. Noble [Trans. Am. Math. Soc. 160, 169–183 (1971; Zbl 0233.54004)] proved that if any power of a space \(X\) is normal, then \(X\) is compact. To be more precise, if \(X^\tau\) is normal for some uncountable \(\tau\geq w(X)\), then \(X\) is compact. Here \(w(X)\) denotes the weight of the space \(X\). As an application of Theorem 1, we obtain the following generalization of Noble’s theorem [loc. cit.].
Theorem 4. If \(X^\tau\) is an \(F_\sigma\)-paranormal space for some uncountable \(\tau\geq w(X)\), then the space \(X\) is compact.
MSC:
| 54B10 | Product spaces in general topology |
| 54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |
Keywords:
normality; paranormality; countable paracompactness; hereditary paranormality; compactness; perfect normality; metrizability; \(F_\sigma\)-paranormalityReferences:
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