Questions tagged [paracompactness]

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Partitions of unity with arbitrary Lip-constants

Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
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1answer
76 views

Is the topology generated by the union of a chain of paracompact topologies paracompact?

Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ ...
1
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1answer
153 views

Is every paracompact topology contained in a maximal paracompact topology?

If $(X,\tau)$ is a paracompact, is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?
4
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1answer
201 views

Paracompactness of Quotient by Group Action

Suppose $X$ is a metric space with a free group action by a topological group $G$, which is also a metric space, such that $\pi\colon X \to X/G$ is a fiber bundle. Does the quotient inherit the ...
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2answers
258 views

How to choose a continuous function which vanishes **only** on the closed set

We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9: Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a ...
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2answers
303 views

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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2answers
246 views

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?
3
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1answer
474 views

CCC + collectionwise normality => paracompact?

Is there a CCC and collectionwise normal space, that isn't paracompact? As we know, CCC + monotone normality => Lindelöf. CCC + collectionwise normality => paracompact? CCC = countable chain ...
3
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1answer
202 views

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 ...
4
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4answers
902 views

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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2answers
2k views

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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2answers
555 views

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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3answers
1k views

Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
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1answer
409 views

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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2answers
854 views

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 ...
3
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1answer
220 views

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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5answers
2k views

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 ...