Questions tagged [paracompactness]
The paracompactness tag has no usage guidance.
23
questions
5
votes
0
answers
109
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 ...
- 790
6
votes
1
answer
115
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 ...
- 4,319
2
votes
1
answer
157
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 $...
- 655
4
votes
1
answer
143
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 ...
- 655
3
votes
1
answer
117
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 ...
1
vote
0
answers
135
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 ...
- 111
1
vote
1
answer
168
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 $\...
- 19
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 $\sigma$ ...
- 43.6k
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?
- 43.6k
4
votes
1
answer
348
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 ...
- 915
1
vote
2
answers
386
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 ...
- 135
2
votes
2
answers
397
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?
0
votes
2
answers
262
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?
- 1,654
3
votes
1
answer
518
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 ...
- 634
3
votes
1
answer
229
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 ...
- 32.9k
4
votes
4
answers
994
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 ...
- 32.9k
25
votes
2
answers
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 ...
- 1,197
2
votes
2
answers
632
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 ...
- 1,331
4
votes
3
answers
2k
views
Paracompact but not Hausdorff
Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
- 15.2k
6
votes
1
answer
433
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 ...
- 1,331
9
votes
2
answers
911
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 ...
- 7,180
3
votes
1
answer
239
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 ...
- 26.7k
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 ...
- 23.5k