# Questions tagged [paracompactness]

The paracompactness tag has no usage guidance.

25
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### The classifying space of any topological group is paracompact and locally contractible

I read somewhere that the classifying space $B_{G}$ for any topological
group $G$ is paracompact and locally contractible. How can I prove this or
can you give me a reference?
Another question that I ...

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### Under what assumption on a proper map does the preimage of sufficiently small neighborhood is homotopy equivalent to the fiber?

Let $\pi\colon X\rightarrow Y$ be a proper map of topological spaces. Let's assume that both $X$ and $Y$ are paracompact, Hausdorff and locally weakly contractible. Then is it enough to conclude that ...

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### For which $X$ is $X\times I$ collectionwise normal?

Many normality-type properties can be characterised in terms of products with the unit interval $I=[0,1]$. For instance, if $X$ is a Hausdorff space, then;
$X$ is normal and countably paracompact if ...

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### A stronger version of paracompactness

Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...

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### When does the refinement of a paracompact topology remain paracompact?

Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$.
Is it true ...

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### Example of locally contractible topological space where Čech cohomology does not coincide with singular cohomology

I believe that it is shown in EH Spanier's "Algebraic Topology" that if 𝑋 is paracompact and locally contractible, then singular cohomology and Čech cohomology of 𝑋 coincide, with ...

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### Relative compactness... but what is the toplogy?

The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...

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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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### 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 $\sigma$ ...

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1
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### 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?

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### 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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### 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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### 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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### 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 ...

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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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### Zariski topology and compact \paracompact space?

Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?

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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 ...