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Questions tagged [paracompactness]

For questions about paracompact spaces as well as variants such as metacompact spaces

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Link between topological dimension and Hamel (algebraic) dimension of a vector space

I was wondering if there is a link between this two dimension definitions in the case of a Topological Vector Space in fact I know that sometimes topological dimension coincides with other notions of ...
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1answer
26 views

Closed Locally Finite Refinement Indexed by Original Cover

Suppose $X$ is a regular, Hausdorff space and that every open cover of $X$ has a locally finite refinement (not necessarily open or closed). Let $\mathcal{U}$ be an open cover of $X$. I want to ...
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Covering characterization of Metrizability and Paracompactness

I am quite struck by the similarity between covering charachterization of Paracompactness and Metrizability both in the T1 and T3 topologies. In particular in the case of T1 topologies we have the ...
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1answer
58 views

Intuition behind Covering Axioms

Many concepts in General Topology are the direct abstraction of very profound and natural concepts (think of structures as topology or uniformity themselves, separation axioms, quotient and ...
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Partitions of unity $\Leftrightarrow$ Hausdorff + Second-countable (in locally Euclidean space)

Let $X$ be a (connected) topological space with a $C^\infty$ atlas. It is a known theorem that if $X$ is second-countable and Hausdorff, then it admits partitions of unity. I'm trying to prove the "...
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A cover of Locally connected space with certain compactness property

Suppose $X$ is a locally connected Hausdorff space. If $X$ is $\sigma$-compact and locally compact, is it always possible to find a countable set of precompact connected open sets $\{U_n\}$ (which ...
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60 views

Locally compact topological group is paracompact

Let $G$ be a locally compact, connected topological group.Show that $G$ is paracompact.
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25 views

Construct Compact Exhaustion using Paracompactness

Let $M$ be a topolgical $n$-manifold. I have to show that there exist a sequence $(K_i)_{i \in \mathbb{N}}$ of compact subspaces $K_i \subset M$ with properties $K_i \subset K_{i+1} $ for all $i \in \...
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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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77 views

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 ...
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34 views

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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1answer
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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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2answers
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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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Why the product of two manifolds is paracompact?

Some authors define a manifold as a paracompact Hausdorff space that is locally Euclidean. Also it is said that a product of two manifolds is a manifold. However, we know that product of a two ...
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1answer
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Paracompactness of the projectified bundle over a paracompact space

Consider a complex rank $n$ Vector Bundle $V \rightarrow X$. It is a standard Argument in the construction of Chern classes to consider the orthogonal complement to the tautological line bundle ...
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1answer
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Topology given by atlas is paracompact

I'm currently reading Jeffrey M. Lee Manifolds and Differential Geometry book. I don't understand a part in the proof of Proposition 1.32. (iii). Proposition 1.32. says: Let $M$ be a set with a $C^...
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2answers
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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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3answers
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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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1answer
283 views

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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1answer
290 views

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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1answer
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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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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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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 ...
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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 ...