MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

As per a recent question of mine, $\omega_1 \times \beta \mathbb{N}$ is not normal. I'm wondering whether there's some sort of "natural" condition that describes when a space has a normal product with $\beta \mathbb{N}$, analagous to Dowker's characterisation that $X$ is countably paracompact iff $X \times [0, 1]$ is normal.

The context here is that I'm looking for something I can weaken to something sensible, as I have a property implied by $X \times \beta \mathbb{N}$ being normal which I am "surprised" isn't an equivalence ($\omega_1$ has this property) and would like to see if I can show its equivalenct to some slightly weaker topological condition.

share|cite|improve this question
Fixed title to remove LaTeX – Harry Gindi Apr 8 '10 at 10:30
I don't follow. How is that a fix? LaTeX works fine in the titles. – David R. MacIver Apr 8 '10 at 14:39
@David: in the past it hasn't for me, but that could be a browser issue – Yemon Choi Apr 8 '10 at 16:26
Strange. I see e.g. the title of the linked post about $\omega_1$ working fine as LaTeX. – David R. MacIver Apr 8 '10 at 16:31
up vote 6 down vote accepted

The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space $K$. This is because every separable compact Hausdorff space is a perfect image of $\beta\mathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $\mathfrak{c}$-paracompact, then $X \times K$ is normal for every compact Hausdorff space $K$ of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space $K$ since these all have weight at most $\mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $\mathfrak{c}$-paracompact if (and only if) $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.

share|cite|improve this answer
(Note: $\mathfrak{c} = 2^{\aleph_0}$.) – François G. Dorais Apr 11 '10 at 19:53
Lovely. Thanks. – David R. MacIver Apr 11 '10 at 20:33

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.