# Questions tagged [paracompactness]

For questions about paracompact spaces and partitions of unity, as well as variants such as metacompact spaces

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### Paracompact Hausdorff Space with Dense Lindelof subset is Lindelof

Let $X$ be a Paracompact Hausdroff space with a dense subset $A$ which is Lindelöf. Then, $X$ is Lindelof I've written down my attenpt below - As per the hint in the problem, as a paracompact $T_2$ ...
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### Question 2 from Bredon's Topology and Geometry page 39

I got stuck on the following question from Bredon Topology and Geometry Chap 1 sec 12. Suppose $X$ is paracompact. For any open subset $U$ of $X \times [0,\infty)$ which contains $X \times \{0\}$ show ...
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### Proof that any metric space has a $\sigma$-locally finite base

A proof proving that the metric topological space $(X,d)$ has a $\sigma$-locally finite base: For every $x\in X$ and $n\in \mathbb{N}$, consider $\{B(x,\frac{1}{n})\}_{x\in X, n\in \mathbb{N}}$, which ...
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### Prove, if $X$ is compact and $Y$ is paracompact, then $X \times Y$ is paracompact (under product topology) [closed]

I was wondering if what I've done is valid. I'm new to topology and relatively new to proof-based math other than linear algebra and a bit of one-dimensional calc. Any help would be appreciated. https:...
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### Why paracompact spaces are required to be Hausdorff

If paracompactness is supposed to be a generalization of compactness. Why is Huasdorfness required in its definition? It seems like it is more a generalization of compact normal spaces. But the name ...
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### confusion in a proof of local finiteness of partitions of unity in CW complexes in John Lee's Introduction to Smooth Manifolds [duplicate]

Suppose $X$ is a CW complex and $X_n$ is the $n$th skeleton. Suppose that for $k=0, \dots n$ we have defined partitions of unity $(\psi_\alpha^k)$ for $X_k$ subordinate to $(U_\alpha^k)$ satisfying ...
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### partition of unity subordinate to an open cover and bump functions

So i was just reading about bump function and i noticed, they always seem to become relevant whenever we consider partitions of unity subordinate to an open cover. The definition of the partition of ...
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### Uncountable product of many copies of $\mathbb{Z}$ is not paracompact

Let $(X,\tau)$ be the product of uncountably many copies of $\mathbb{Z}$. Prove that $(X,\tau)$ is not paracompact. In order to prove that something is not paracompact. We need to find an open cover ...
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### Every $F_\sigma$-set in a paracompact space is paracompact.

Every $F_\sigma$-set in a paracompact space is paracompact. Definitions: $F_\sigma$-set is a countable union of closed sets paracompact: if every open cover has an open refinement that is locally ...
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### Proof a theorem about Metrizable manifold

Where could the proof of the following theorem be found : a manifold is metrizable if and only if it is paracompact
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### Tietze extension theorem for vector bundles on paracompact spaces

In Atiyah's K-theory book, he gives a proof of the Tietze extension theorem to vector bundles. He gives the proof only for compact Hausdorff spaces, but I want to generalise it to paracompact ...
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### Theorem 41.7 in Munkres Topology

The only part I am having difficulty justifying is why there exists a $W_x \in \{W_\alpha\}$ that intersects only finitely many sets in $\{\mbox{Supp } \psi_\alpha \}$
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### Space of Sequences with Finitely Many Nonzero Terms is Paracompact

I just proved the following theorem: If $X$ is a regular space that can be written as a countable union of compact subspaces of $X$, then $X$ is paracompact. I am now working on the following: ...
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### normal doesn't imply paracompact

I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact. Thanks in advance.
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### Non-trivial explicit example of a partition of unity

Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
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### Are locally contractible spaces hereditarily paracompact?

The question title says it all. For the record, I have no reason to believe that this is true, but my question has a bit of a background. I am reading Ramanan's Global Calculus book because I am ...
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### Product of paracompact spaces

I know that the product of a compact space and a paracompact space is paracompact, and that in general the product of two paracompact spaces are not paracompact. Question: Is there a weakest ...
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### Are subspaces of paracompact spaces normal?

Are all subspaces of a paracompact space normal? This is what I think about this question... First a paracompact Hausdorff space turns out to be Normal, second the paracompact property is not ...
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### Are countably compactly generated spaces paracompact?

A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces. Are countably compactly generated spaces paracompact spaces? Do we have ...
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### Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)

I have a question about a proof in John Lee's Introduction to Topological Manifolds (5.22). Given CW-complex $X$ with skeletons $X_n$ and open cover $\left(U_\alpha\right)_{\alpha\in A}$, we ...
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### Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
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### Why is every one dimensional Complex Manifold paracompact?

I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue ...
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### Paracompactness of CW complexes (rather long)

I finished reading Lee's 'introduction to topological manifolds' (2nd edition) and I'm currently tying up some loose ends. One thing I can't understand is the proof of paracompactness of CW complexes. ...