Normal x Compact Needs Not be Normal

This is to present a non-normal cross product where one factor is a normal space and the other factor is a compact space. The normal space is $X=[0,\omega_1)$ and the compact space is $Y=[0,\omega_1]$ . I would like to show that $X \times Y$ is not normal. For a basic discussion on these ordinal spaces with the order topology, see this previous post.

In addition to demonstrating an example meeting the criteria set forth in the title, I would like to look at this through two angles. Because these spaces have the ordered topology, they are hereditarily normal (completely normal). Note that $X \times Y$ is a subspace of $Y \times Y$ . Thus $Y \times Y$ fails to be hereditarily normal, demonstrating that hereditarily normal x hereditarily normal needs not be hereditarily normal. Another example of this is the double arrow space. In this previous post, I showed that the square of the double arrow space is not hereditarily normal. The double arrow space is perfectly normal and is thus a stronger example in this regard since it shows the square of a perfectly normal space needs not be hereditarily normal. But the uncountable ordinal spaces of $[0,\omega_1)$ and $[0,\omega_1]$ are important counterexamples as well as building blocks for other counterexamples. So for the record, I present them here.

For the second angle, these two spaces show that hereditarily normality is not “nice” enough to prevent non-normal product.

In proving $X \times Y$ is not normal, the Pressing Down Lemma is used. A subset $S$ of $X=[0,\omega_1)$ is a stationary set if $S$ meets every closed and unbounded set in $X$ . The following is one version of the Pressing Down Lemma (see Lemma 6.15 on p. 80 of [1] for a more general version).

Pressing Down Lemma

$S$

$[0,\omega_1)$

$f:S \rightarrow \omega_1$

$\gamma \in S$

$f(\gamma)<\gamma$

$\alpha<\omega_1$

$f^{-1}\lbrace{\alpha}\rbrace$

$[0,\omega_1)$

To show that $X \times Y$ is not normal, let $H$ and $K$ be the following sets.

$H=\lbrace{(\alpha,\alpha):\alpha<\omega_1}\rbrace$

$K=\lbrace{(\alpha,\omega_1):\alpha<\omega_1}\rbrace$

These two sets are disjoint and closed in $X \times Y$ . Let $U$ and $V$ be open such that $H \subset U$ and $K \subset V$ . It follows that $U \cap V \neq \phi$ .

For each $\alpha<\omega_1$ , let $g(\alpha)<\omega_1$ such that $[g(\alpha),\alpha] \times [g(\alpha),\alpha] \subset U$ . Since $g$ is a pressing down function, there is some $\delta<\omega_1$ and there is a stationary set $S$ such that all points in $S$ are mapped to $\delta$ by $g$ . Specifically, for each $\alpha \in S$ , we have: