### On $\alpha $-normal and $\beta $-normal spaces

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...