Is the product of a $T_4$ space and a compact $T_4$ space necessarily $T_4$? Let $N$ be an arbitrary $T_4$ topological space and let $C$ be a compact $T_4$ space.  Is $N\times C$ (with the usual product topology) necessarily $T_4$?
I know that the product of general $T_4$ spaces need not be $T_4$.  For example, the Sorgenfrey line is $T_4$, but the product of the Sorgenfrey line with itself is not $T_4$.  The product of two compact $T_4$ spaces, however, is $T_4$.  This follows from the fact that a $T_4$ space is $T_3$, the product of two $T_3$ spaces is $T_3$, the product of two compact spaces is compact, and a compact $T_3$ space is $T_4$.
What I'm wondering is if the product of a general $T_4$ space with a compact $T_4$ space is always $T_4$.  My intuition says "yes" since compact spaces often act like a single point, but this isn't always so, and intuition can be wrong.  I have no idea how to formulate a proof and I can't think of an obvious counter-example.  For example, the product of the Sorgenfrey line with a compact $T_4$ space will be $T_4$, since the Sorgenfrey line is paracompact, the product of a paracompact space with a compact space is paracompact, and a paracompact $T_3$ space is $T_4$.  So can anyone provide or give a reference to a proof or counter-example to my question?  
 A: For any $T_4$ space $X$ , $X \times \beta X$ ($\beta X$ is the Cech-Stone compactification of $X$ which is defined for all Tychonoff ($T_{3\frac12}$) spaces) is normal iff $X$ is paracompact. This is due to Tamano (1960). So 
the paracompactness strengthening of normality is essential to get normality with a product with any compactum. Otherwise put, if $X$ is normal but not paracompact,
then there is at least one compact space $C$, namely $C=\beta X$, such that $X \times C$ is not normal.
For a link to the original paper and a direct proof by Scott that $\omega_1 \times (\omega_1 +1)$ is not normal (a special case) see this thread. The last special case is quite strong as then we have a normal $N = \omega_1$ which is hereditarily normal and locally compact, first countable so quite nice, and $C = \omega_1 +1$ is compact, and also hereditarily normal.
For a positive result, if we only want the product with $[0,1]$ to be normal, then Dowker proved that for normal $X$, $X \times [0,1]$ is normal iff $X$ is countably paracompact. In ZFC there are (quite rare) so-called Dowker spaces that are normal and not countably paracompact (so that $X \times [0,1]$ is not normal). Ordered spaces are countably paracompact so $\omega_1 \times [0,1]$ is normal, while $\omega_1 \times (\omega_1 +1)$ is not. So sometimes it depends on the compact space.
