As per [a recent question of mine](https://mathoverflow.net/questions/20228/omega-1-times-beta-mathbbn-normal), $\omega\_1 \times \beta \mathbb{N}$ 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.



Is there a "natural" characterization of when X × βN is normal?
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*](http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50119.pdf), MR[132525](http://www.ams.org/mathscinet-getitem?mr=132525)), 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.


