product of an arbitrary topological space with a compact Hausdorff space

What can we say about the product space (with product topology) $X\times Y$ if $X$ is an arbitrary topological space and $Y$ is a compact Hausdorff space?

Do we know from the given information if the projection $\pi:X\times Y\to X$ is open/closed?

Regards

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The projections for a product space are always open. –  Dylan Moreland Feb 2 '12 at 2:41
–  Jonas Meyer Feb 2 '12 at 2:44

1 Answer

The product topology is defined in such a way that the projection maps are open. A projection map parallel to a compact factor is always closed; the factor need not be Hausdorff. This is a standard basic result in topology; you can find a proof in the first paragraph of this answer.

Beyond that very little can be said about $X\times Y$.

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