Abstract: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. \par 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 Kat\v etov'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).
Keywords: normal, $\alpha $-normal, $\beta $-normal, $\kappa $-normal, weakly normal, extremally disconnected, $C_p(X)$, Lindel\"of, compact, pseudocompact, countably compact, hereditarily separable, hereditarily $\alpha $-normal, property $wD$, weakly perfect, first countable
AMS Subject Classification: 54D15, 54D65, 54G20