Questions tagged [paracompactness]

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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 ...
Cary's user avatar
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11 votes
5 answers
3k 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 ...
Anton Geraschenko's user avatar
9 votes
2 answers
925 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 ...
Vipul Naik's user avatar
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6 votes
1 answer
435 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 ...
David R. MacIver's user avatar
6 votes
1 answer
123 views

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 ...
Tyrone's user avatar
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5 votes
2 answers
339 views

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 ...
Mehmet Onat's user avatar
  • 1,161
5 votes
0 answers
120 views

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 ...
user42024's user avatar
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4 votes
4 answers
1k 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 ...
David Roberts's user avatar
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4 votes
3 answers
2k views

Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
David Carchedi's user avatar
4 votes
1 answer
146 views

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 ...
Cla's user avatar
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4 votes
1 answer
364 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 ...
ThorbenK's user avatar
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3 votes
1 answer
524 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 ...
Paul's user avatar
  • 634
3 votes
1 answer
240 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 ...
Chris Schommer-Pries's user avatar
3 votes
1 answer
138 views

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 ...
Joel Springer's user avatar
3 votes
1 answer
233 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 ...
David Roberts's user avatar
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2 votes
2 answers
333 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?
Fred Dashiell's user avatar
2 votes
2 answers
412 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?
Guilherme Carmona's user avatar
2 votes
1 answer
178 views

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 $...
Cla's user avatar
  • 755
2 votes
1 answer
91 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 $\sigma$ ...
Dominic van der Zypen's user avatar
2 votes
2 answers
659 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 ...
David R. MacIver's user avatar
1 vote
1 answer
161 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?
Dominic van der Zypen's user avatar
1 vote
2 answers
438 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 ...
C. Ding's user avatar
  • 135
1 vote
0 answers
136 views

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 ...
edamondo's user avatar
  • 111
1 vote
1 answer
185 views

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 $\...
user149610's user avatar