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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 ...
mikeqwertyuiop's user avatar
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1 answer
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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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If the open cover U of the topological space has a partition of unity then U has a locally finite refinement.

In the of Engelking Book "General Topology in the chapter of paracompactness there is a lemma related to partition of unity and locally finite refinement. In the lemma it is given that $$f_{s}(x)&...
Haxhi Dacaj's user avatar
3 votes
1 answer
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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 ...
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Example of a locally finite refinement of a family

Consider $C$ is the open cover { $(-n,n): n$$\in$ lN}. I would like to cconstruct a locally refinement of $C$. What I considered was $C'$={ $(n,n+2): n$$\in$ lN}. This covers the real line, is locally ...
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Paracompactness properties of the deleted Tychonoff plank

I am looking for a justification of paracompactness and related properties for the deleted Tychonoff plank. The deleted Tychonoff plank is the space $X=((\omega_1+1)\times(\omega+1))\setminus\{\langle\...
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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 ...
PatrickR's user avatar
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4 votes
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Union of two open paracompact subspaces is paracompact?

Is the union of two open paracompact subspaces of a space $X$ paracompact? A space is called paracompact if every open cover of the space has a locally finite open refinement. Proof attempt: Suppose $...
PatrickR's user avatar
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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 ...
Jakobian's user avatar
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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'...
math-physicist's user avatar
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Paracompactness of closed subspaces

Suppose that $X$ is a paracompact topological space and let $S \subseteq X$ be a closed subspace. It is well-known that $S$ itself is paracompact, when equipped with the subspace topology. But what ...
shuhalo's user avatar
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Open "enlargement" of locally finite family of compact sets

I want to show that Let $M$ be a paracompact $T_2$ space and let $(K_i)_{i\in I}$ be a locally finite family of compact subsets of $M$, then there is a locally finite open covering $(U_j)_J$ of $M$ ...
William von Schwarz's user avatar
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Product of $F_{\sigma}$-screenable spaces

Does anyone know if the product $X \times X$ is $F_{\sigma}$-screenable when the space $X$ is $F_{\sigma}$-screenable ?
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A problem about paracompact Hausdorff space and its closed subset.

Let X be a paracompact Hausdorff space. $M=\bigcup_{i=1}^{\infty}F_i$, $F_i$ is closed. Prove that M is paracompact as a subspace of X. Idea: In Munkres' book topology, there are two theorems: (1) ...
save123's user avatar
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Prove that if $f:X\longrightarrow Y$ is closed and continuous then $Y$ is paracompact (and hausdorff) provided that $X$ is.

I was studying paracompactness by Munkres topology text when I ran into the following exercise Prove that paracompactness is an invariant with respect perfect maps defined in hausdorff spaces. which ...
Antonio Maria Di Mauro's user avatar
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$\mathbb{R}_{\rm Sorg.}$ is paracompact but $\mathbb{R}_{\rm Sorg.} \times \mathbb{R}_{\rm Sorg.}$ is not

I’m trying to solve this problem: $\def\bbR{\mathbb{R}} \def\RSorg{\bbR_{\rm Sorg.}} \def\calR{\mathcal{R}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \DeclareMathOperator{\range}{range}$ Prove ...
Paul's user avatar
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Paracompact but not Lindelöf and vice versa

We know that a regular Lindelöf space is paracompact. Can anybody give me an example where the space is Lindelöf but not paracompact ? I have found that $\mathbb{R}$ with discrete topology is ...
nkh99's user avatar
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Is the Mysior Space paracompact?

I want to find out if the space defined here is paracompact. I think it is not but I'm not able to prove it. I tried applying the fact that a Lindelöf T3 Space is Paracompact but this space isn't ...
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compactly generated and paracompact $\Rightarrow$ Hausdorff?

In A Concise Course in Algebraic Topology by May, a proposition is stated that any open cover of a paracompact space has a numerable refinement, where the space is assumed to be compactly generated ...
LuckyJollyMoments's user avatar
3 votes
1 answer
264 views

Concerning topological manifolds: Are paracompact and connected locally euclidian Hausdorff spaces always second-countable?

There are different definitions for topological manifolds, sometimes second-countability or paracompactness are added to being locally euclidian Hausdorff. (Sometimes Hausdorff is also left out, but I ...
Samuel Adrian Antz's user avatar
2 votes
2 answers
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Finite union of closed paracompact subspaces is paracompact.

This is an excercise from the Munkres, Section 41, excercise 7)a: If $X$ is regular and is a finite union of closed paracompact subspaces of $X$, then $X$ is paracompact. Well, I tried to prove it for ...
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Motivation behind the definition of paracompact.

I am self-studying topology.I encountered the definition of a paracompact space.A collection $\mathcal A$ of subsets of $X$ is said to be locally finite if each point in $X$ has an open neighborhood ...
Kishalay Sarkar's user avatar
3 votes
1 answer
138 views

Is every open set in a subspace the intersection of the subspace with an open set "having the right closure"?

Let $X$ be a Hausdorff paracompact topological space and $Y ⊂ X$ closed. By definition every open $V ⊂ Y$ is the intersection of $Y$ with an open set $U ⊂ X$. However for such an $U$ we are not ...
Carlos Esparza's user avatar
1 vote
1 answer
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Are these two definitions for paracompactness equivalent at least for manifolds?

My lecturer defined paracompactness as follows: A topological space $X$ is called paracompact if for any open cover $\{U_\alpha\}$ of $X$ there is a locally finite subcover, i.e. for any $x\in X$ ...
MathIsCool's user avatar
3 votes
1 answer
2k views

Every metric space is paracompact (an elegant proof)

I'm studying a different proof to show that each metric space is paracompact in the book: Singh, Tej Bahadur-Introduction to Topology. It is a very elegant construction unlike the inductive method ...
Inquirer's user avatar
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Does locally compact and paracompact imply $X\setminus E$ is contained in a countable union of compact sets, when $E$ has finite (Baire) measure?

I have been trying to get an understandable, self-contained proof about the regularity of Baire measures in locally compact spaces. Failing to follow Royden's proof - which required the lemma I asked ...
FShrike's user avatar
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Every closed subspace of a paracompact space $X$ is paracompact.

Every closed subspace of a paracompact space $X$ is paracompact. My attempt: Let $A\subset X$ be closed and $\{U_{\alpha}\}_{\alpha \in I}$ an open cover of $A$. This means that $U_\alpha = A \cap U_{\...
Inquirer's user avatar
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1 answer
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How Regularity of $X$ implies $\mathcal {A}$ covers $X$?

[Reference:Munkres Topology $2nd$ edition] $X$ is PARACOMPACT+ $X$ is HAUSDORFF$\implies $ $X$ is NORMAL$\implies $ $X$ is Regular. I'm not getting (1)How does "Regularity of X implies that $\...
Styles's user avatar
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1 vote
1 answer
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Is the deleted total space of a vector bundle over a paracompact space paracompact?

In the introductory book by Milnor and Stasheff on characteristic classes, to define the Chern classes of some vector bundle $\varphi : E \to B$, we construct a vector bundle over the total deleted ...
QuinnLesquimau's user avatar
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2 answers
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In a $T_3$-space with $\sigma$-locally finite base, every open set is an $F_\sigma$ set.

I am trying to understand the last line of this proof. Why is the union of all $c(B_k)$ equal to $G$? I dont understand the difference between $B_{n(x),\lambda(x)}$ and $B_{k,\lambda(x)}$. I know that ...
gbd's user avatar
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Product of paracompact and compact spaces [duplicate]

“If $X$ is paracompact and $Y$ is compact, then $X \times Y $ is also paracompact.” From this , if $X$ is a discrete space and $Y = \{0, 1\}$ with the topology $\{\emptyset, Y, \{0\}\}$, then the ...
Marwa Zeyad's user avatar
3 votes
1 answer
54 views

countably locally finite and locally finite covers

I am trying to understand the above proof. My problem is that I fail to understand why we need to construct the neighborhood $W_1\cap...\cap W_n\cap H_x$ of $x$ that interescts with finite many of $\...
user3741635's user avatar
1 vote
1 answer
136 views

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. ...
gbd's user avatar
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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 ...
user40102's user avatar
2 votes
1 answer
639 views

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 ...
José Carlos Santos's user avatar
1 vote
1 answer
72 views

If a locally finite open cover has a locally finite closed refinement, it has a locally finite open barycentric refinement

If a locally finite open cover has locally finite closed refinement, does it have a locally finite open barycentric refinement? We can get an open barycentric refinement like this. Let $(U_i)_{i \in I}...
Ris's user avatar
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1 answer
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If $X$ is paracompact then $f(X)$ not need to be paracompact

If $X$ is paracompact and $f:X \to Y$ is contínuous then $f(X)$ need not to be paracompact. I've found this exercise at Munkres. My effort is the following: I've proved that every discrete space is ...
Joãonani's user avatar
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5 votes
1 answer
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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 $\mathbb{R}^n$, ...
Ali Pedram's user avatar
2 votes
1 answer
724 views

An Example for the Partition of Unity for a Circle

I am trying to wrap my head around on what partition of unity means and trying to understand some examples of it. I have an engineering background so most of the abstract discussions are a little hard ...
Ali Pedram's user avatar
2 votes
0 answers
71 views

Is the product of paracompact with $\sigma$-compact always paracompact?

Exercise A space $X$ is said to be $\sigma$-compact if it can be written as a countable union of compact subspaces. Is the product of a paracompact space $X$ with a $\sigma$-compact space $Y$ always ...
Daniel Kawai's user avatar
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4 votes
2 answers
214 views

Paracompactness of topological group.

If $G$ be a locally compact topological group. Show that $G$ is paracompact. Note: If we restrict $G$ to be locally compact, connected topological group, this problem becomes easier by constructing a ...
Hoang Nguyen's user avatar
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2 answers
163 views

show that every countable subcover is $\sigma$- locally finite refinement. (every regular Lindelöf space is paracompact)

I was trying to show that the proposition above, if a topological space $X$ is regular (which I mean seperating the points and closed sets) and Lindelöf, then the space is paracompact. For any open ...
Dans0804's user avatar
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1 answer
28 views

locally finite family $ \mathcal{A} $

Hello I have had problems with this exercise Let $X$ be a topological space and $ \mathcal{A} \subset \mathcal{P}(X) $ a locally finite family such that for any $ A, B \in A $ with $ A \neq B $ and $ \...
Pokeamigo 6's user avatar
1 vote
1 answer
111 views

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 ...
Sophie M's user avatar
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1 answer
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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. ...
D.R.'s user avatar
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0 votes
1 answer
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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 $\mathcal{V}$...
Emo's user avatar
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1 vote
1 answer
159 views

$\mathbb R^{\omega}$ 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). ...
Sphere's user avatar
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0 answers
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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 ...
Sphere's user avatar
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1 vote
1 answer
281 views

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. ...
Sphere's user avatar
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1 answer
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If for an open cover there exists a partition of unity subordinated to it, then the cover has an open locally finite refinement.

In the Engelkings book "General Topology" in the chapter of paracompactness there is a lemma related to partitions of unity and locally finite covers. I am trying to understand the proof of ...
Emo's user avatar
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