# Questions tagged [paracompactness]

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

27 questions
1answer
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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 ...
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 ...
1answer
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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 ...
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 ...
2answers
62 views

### 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 "...
1answer
31 views

### 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 ...
1answer
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### Locally compact topological group is paracompact

Let $G$ be a locally compact, connected topological group.Show that $G$ is paracompact.
1answer
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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?
3answers
303 views

### 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 ...
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 ...
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 ...
1answer
107 views

### 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 ...
1answer
244 views

### 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 ...
0answers
203 views

### 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 ...
1answer
346 views

### 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 ...
3answers
2k views

### 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 ...