# Questions tagged [paracompactness]

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

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### Each elements in Locally finite precompact open cover of a topological manifold has at most finite intersection with others

The problem is from John M Lee's Introduction to Smooth Manifold:problem 1.4.(see below image) I have checked similar post regarding this problem from elsewhere of this site. However, I still find it ...
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### Help in understanding paracompactness

I have a great troubles in understanding the definition of paracompactness. I tried to make up some examples to better understand the definition. There is a theorem that says that every topological ...
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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 ...
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### Why is paracompactness needed to prove a regular space is normal?

I am trying to understand why paracompactness is needed to prove a regular space is normal. It was used in the prove above to find a locally finite open refinment $\{w_\lambda\}$ from an open cover. ...
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### Intersection of Open Set and Complement of Compact Set Is Open

I am reviewing the blog post: https://amathew.wordpress.com/2010/08/17/paracompactness. Under Lemma 7, the author states that each $U_{i+1} - \overline{U_{i-2}}$ is open in $X$ (which is locally ...
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### An example of a non-paracompact topological space

The most famous example of a non-paracompact space is the long line. I saw in several places a reference to another example: the Cartesian product of uncountably many copies of an infinite discrete ...
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### Help with proof of Lemma 1 in "A note on paracompact spaces" (Michael, PAMS 4 (5), 1953)

In the paper by Ernest Michael cited in the title of this question, the following Lemma is crucial, but a step in the proof confuses me. Lemma 1. Let $X$ be a regular space. Then the following are ...
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### Why do we define partitions of unity to be locally finite instead of pointwise finite?

The typical definition of a partitions of unity $\{\varphi_i\}_{i\in I}$ subordinate to a cover $\{U_i\}_{i\in I}$ requires that for any $x\in X$, there is a neighborhood $N_x$ containing $x$ s.t. ...
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### Does normal space guarantees existence of normal cover

Let $X$ be a topological space and $\mathcal{U}$ be an open cover of $X$. Define $st(M)=\{ U\in\mathcal{U}\vert M\cap U \neq \emptyset \}$ where $M$ is an arbitrary subset of $X$. A cover $\mathcal{V}$...
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### $\mathbb R^{\omega}$ in the box topology not paracompact?

I know that we can't prove that $\mathbb R^{\omega}$ in the box topology is normal, so we can't say for sure that it is paracompact(it is Hausdorff, and every paracompact Hausdorff space is normal). ...
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### Perfect image of a metrizable space is metrizable.

How can I prove that the perfect image of a metrizable space is metrizable? I know the following three theorems about equivalent conditions of metrizability. A space $X$ is metrizable if and only if ...
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1 vote
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### countable union of closed paracompact subspaces

This is an exercise of Munkres topology section 41. Let $X$ be a regular space. If $X$ is a countable union of closed paracompact subspaces of $X$ whose interiors cover $X$, show $X$ is paracompact. ...
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