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normal; $\alpha$-normal; $\beta$-normal; $\kappa$-normal; weakly normal; extremally disconnected; $C_p(X)$; Lindelöf; compact; pseudocompact; countably compact; hereditarily separable; hereditarily $\alpha $-normal; property $wD$; weakly perfect; first countable
We define two natural normality type properties, $\alpha$-normality and $\beta$-normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every $\beta$-normal, pseudocompact space is countably compact, and show that if $X$ is a dense subspace of a product of metrizable spaces, then $X$ is normal if and only if $X$ is $\beta$-normal. All hereditarily separable spaces are $\alpha $-normal. A space is normal if and only if it is $\kappa$-normal and $\beta$-normal. Central results of the paper are contained in Sections 3 and 4. Several examples are given, including an example (identified by R.Z. Buzyakova) of an $\alpha$-normal, $\kappa $-normal, and not $\beta$-normal space, which is, in fact, a pseudocompact topological group. We observe that under CH there exists a locally compact Hausdorff hereditarily $\alpha $-normal non-normal space (Theorem 3.3). This example is related to the main result of Section 4, which is a version of the famous Katětov's theorem on metrizability of a compactum the third power of which is hereditarily normal (Corollary 4.3). We also present a Tychonoff space $X$ such that no dense subspace of $X$ is $\alpha $-normal (Section 3).
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