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Questions tagged [paracompactness]

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

A topological space is called paracompact if every open cover has a locally finite refinement. Some authors also require paracompact spaces to be Hausdorff.

Most applications of paracompactness arise from the fact that a paracompact Hausdorff space admits a partition of unity subordinate to any open cover. Partitions of unity are useful for "gluing" local structure on a space to obtain a global structure. Manifolds are commonly assumed to be paracompact so that they have partitions of unity.

This tag can also be used for questions on variants of paracompactness, such as metacompactness (every open cover has a point-finite refinement) or countable paracompactness (every countable open cover has a locally finite refinement).

130 questions
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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. The proof contains some mistakes I feel (perhaps…
user58664
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The product of a paracompact space and a compact space is paracompact. (Why?)

A paracompact space is a space in which every open cover has a locally finite refinement. A compact space is a space in which every open cover has a finite subcover. Why must the product of a compact and a paracompact space be paracompact? I really…
Mark
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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 since paracompact spaces are collection-wise…
chrystomath
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Is the Hausdorff assumption missing in Hatcher Proposition 4G.2?

Below is an image of Hatcher Proposition 4G.2, which is used to prove the Nerve Theorem (Hatcher Corollary 4G.3). My question is about the sentence that is highlighted in yellow. I thought that the existence of a partition of unity subordinate to…
R. H. Vellstra
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Finite Partitions of Unity in a Compact Hausdorff Space

I'm working on this proof in Gamelin "Introduction to Topology" and I think I'm almost at the result, I'm just a little stuck with how to proceed. It is this. Let $X$ be a be compact Hausdorff space and let $\{U_\alpha\}_{\alpha \in A}$ be an open…
Leif Ericson
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Paracompact and Compactly Generated spaces

A couple of days ago, thanks to Strom's excellent book Modern Classical Homotopy Theory, I started reading up on compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces (the best decision in my life so far).…
Olivier Bégassat
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Fully normal implies paracompact without a $T_1$ assumption?

It's well-known that a $T_1$ topological space is fully normal if and only if it is $T_2$ and paracompact. It appears, looking at the proofs from Henno Brandsma's nice exposition here and here, that we can drop the $T_1$ assumption for the…
M W
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Paracompactness properties of the line with multiple origins

Let $X$ be the "line with multiple origins", obtained by taking a set $S$ with the discrete topology and taking the quotient space of $\mathbb R\times S$ by the equivalence relation that identifies $(x,\alpha)$ with $(x,\beta)$ whenever $x\ne 0$. …
PatrickR
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Every paracompact space with the Suslin property is Lindelöf

I've been reviewing exams, and I came across this problem that I am having trouble with. Can anyone help me out? Show that every paracompact space with the Suslin property is Lindelöf. A topological space has the Suslin property if there is no…
josh
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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 inductively define partitions of unity…
Marcin Łoś
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Is every paracompact CCC space Lindelöf?

In Every paracompact space with the Suslin property is Lindelöf and How to prove that every Paracompact space with the Suslin property is Lindelof it's observed that every preregular paracompact space with the Suslin property, a.k.a. the countable…
Steven Clontz
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Examples of spaces that are $T_5$ but aren't countably paracompact.

After reading that every perfectly normal space is countably paracompact, I wanted to find an example of a space that is $T_5$, but isn't countably paracompact. However, no examples of spaces verifying this can be found in $\pi$-Base yet. I have…
Almanzoris
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Is Bing's discrete extension space realcompact?

See here for definition of Bing's space. There's also Dan Ma's blog or Counterexamples in Topology by Steen and Sebach. Since the Michael's closed subspace $Y\subseteq X$ of Bing's space $X$ is metacompact but not paracompact, $Y$ is not countably…
Jakobian
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A (short) proof for the paracompactness of CW complexes

So, for a while now I've been looking for a short but concise proof of the fact that Every CW complex is paracompact. I finally found the following proof (shortest so far) of this theorem but couldn't understand a few things the author did…
math-physicist
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An Intuition for Paracompactness

I do have an intuitive understanding of compactness based on Euclidean space but not so much for paracompactness. Based on the Heine-Borel theorem, for a subset $S$ of Euclidean space $\mathbb{R}^n$, the following two statements are equivalent: $S$…
Ali Pedram
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