Questions tagged [paracompactness]

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

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Paracompact Hausdorff Space with Dense Lindelof subset is Lindelof

Let $X$ be a Paracompact Hausdroff space with a dense subset $A$ which is Lindelöf. Then, $X$ is Lindelof I've written down my attenpt below - As per the hint in the problem, as a paracompact $T_2$ ...
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Question 2 from Bredon's Topology and Geometry page 39

I got stuck on the following question from Bredon Topology and Geometry Chap 1 sec 12. Suppose $X$ is paracompact. For any open subset $U$ of $X \times [0,\infty)$ which contains $X \times \{0\}$ show ...
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Proof that any metric space has a $\sigma$-locally finite base

A proof proving that the metric topological space $(X,d)$ has a $\sigma$-locally finite base: For every $x\in X$ and $n\in \mathbb{N}$, consider $\{B(x,\frac{1}{n})\}_{x\in X, n\in \mathbb{N}}$, which ...
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47 views

Prove, if $X$ is compact and $Y$ is paracompact, then $X \times Y$ is paracompact (under product topology) [closed]

I was wondering if what I've done is valid. I'm new to topology and relatively new to proof-based math other than linear algebra and a bit of one-dimensional calc. Any help would be appreciated. https:...
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34 views

Why paracompact spaces are required to be Hausdorff

If paracompactness is supposed to be a generalization of compactness. Why is Huasdorfness required in its definition? It seems like it is more a generalization of compact normal spaces. But the name ...
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18 views

Paracompactness of Adjunction Spaces

In this question I will understand the definition of paracompactness to include the Hausdorff separation axiom. Let $X$ be a paracompact space and $A\subseteq X$ a closed subspace. Let $f:A\...
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43 views

Continuously summing a family of maps.

Let $X$ be a space and $$\{f_i:X\rightarrow[0,\infty)\}_{i\in\mathcal{I}}$$ a family of continuous maps $X\rightarrow [0,\infty)$ indexed by some set $\mathcal{I}$. Assume that the family is point-...
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50 views

Even covers in Kelley's General Topology

Kelley in his book General Topology introduces a notion of even cover. A cover $\mathcal{U}$ of a topological space $X$ is even if there exists a neighborhood $\mathcal{V} \subset X \times X$ of the ...
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27 views

Doubt on paracompactness and lorentz signature.

While the study of Manifolds are quite interresting by itself, we know that General Relativity have on it's core the study of Manifolds. When we transport the Manifold mathematical structure to ...
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37 views

Existence of partition of unity such that the functions and their derivatives vanish at infinity

Like the title says, suppose $K\subset \mathbb{R}$ is a locally compact space without isolated points and $\left \{ U_i \right \}$, ${i \in \{1,\cdots ,n \}}$ is an open cover of $K$. Are There a ...
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22 views

Extension of sections from an open subset of closed set

Suppose $X$ is a locally compact and paracompact topological space. Let $Y$ be a closed subset of $X$, let $i:Y \to X$ be an embedding and $\mathcal{F}$ sheaf on $X$. I want to show that for any open ...
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33 views

Use partition of unity to show existence of positive integral on the whole space

Let $X$ be a paracompact Hausdorff space and let $C_c(X)$ be the set of all continuous real-valued functions on $X$ with compact support. By a positive integral on $X$ I will mean a linear function $I:...
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1answer
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confusion in a proof of local finiteness of partitions of unity in CW complexes in John Lee's Introduction to Smooth Manifolds [duplicate]

Suppose $X$ is a CW complex and $X_n$ is the $n$th skeleton. Suppose that for $k=0, \dots n$ we have defined partitions of unity $(\psi_\alpha^k)$ for $X_k$ subordinate to $(U_\alpha^k)$ satisfying ...
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partition of unity subordinate to an open cover and bump functions

So i was just reading about bump function and i noticed, they always seem to become relevant whenever we consider partitions of unity subordinate to an open cover. The definition of the partition of ...
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To show that if every open set of a topological space X is paracompact, then every set in X is paracompact.

I want to show that if every open set of a topological space $X$ is paracompact, then every set in $X$ is paracompact. My idea was to first take an arbitrary set $A \subseteq X$ and an open conver $\{...
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1answer
38 views

Countable union of paracompact spaces is Paracompact with regularity?

When studying "Paracompactness", I thought that If $X$ is regular, and $X= \bigcup Y_n$ ($n\ge1$) where $Y_n$ is paracompact subspace of $X$, then $X$ is paracompact. Here is the proof. ...
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52 views

Product of Paracompact spaces being Paracompact

I'm interested in the game-characterization proposed by Telgarsky (paper) of the class of paracompact spaces that preserve paracompactness under cartesian product with another paracompact space. He ...
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61 views

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
51 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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1answer
50 views

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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137 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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177 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 "...
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41 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 ...
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152 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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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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42 views

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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1answer
197 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 locally ...
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44 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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1answer
78 views

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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39 views

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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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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A property of neighborhoods of the diagonal $\Delta\subset Y\times Y$ obtained from a paracompact space $Y$

Let $Y$ be a topological space and $V\subset Y\times Y$. For each $y\in Y$, we define $$V[y]=\{z\in Y\,:\,(y,z)\in V\}.$$ It is possible to show that, if $U\subset Y\times Y$ is an open set and $\...
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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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118 views

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
210 views

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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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
387 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 ...
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1answer
460 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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335 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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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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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 ...
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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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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. ...
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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 ...