The paracompactness tag has no usage guidance.

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### Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?

This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?

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### Does locally compact plus pseudocompact imply paracompact?

This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?

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### Characterisation of paracompact spaces by some sort of embeddability?

This question was inspired by this question.
Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another ...

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### An example of a non-paracompact tvs (over the reals, say)

What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...

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### CW complexes and paracompactness

It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two ...

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### Conditions useful for proving paracompactness

I have a family of properties which I want to show taken together imply paracompactness (I can show that they are all implied by paracompactness). I can prove a whole bunch of things which are ...

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### Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.

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### Countable paracompactness, normality and locally countable open covers

(repost from the topology Q&A board)
I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x ...

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### Space whose product with paracompact space is paracompact

Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?
All compact spaces have this property (this can be shown from the ...

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### Are mapping spaces paracompact?

Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and ...

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### Is the long line paracompact?

A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...