Questions tagged [compactness]
The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness. This includes logical compactness.
6,564 questions
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Order in an open set
I need help to prove a theorem.
In the following definition $\operatorname{fr}_X(A)$ means the boundary of $A$ in $X$
Definition. Let $(X, \tau_X)$ be a topological space, let $p \in X$, and let $\...
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Is every CGWH space sober?
Definition 1: A topological space is called compactly generated if it has the following property: Let $U \subseteq X$ be a subset such that for every compact Hausdorff space $K$ and every continuous ...
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Compact operators in Hilbert space
I have tried to solve this exercise:
Let $H$ be a Hilbert space and let $T = T^* \in B_{\infty}(H)$ a compact operator. Given $\psi_0 \in H$ consider the equations:
\begin{align*}
(1)\quad &T\psi =...
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Is the quotient of the subspace homeomorphic to the subspace of the quotient?
Suppose $X$ is compact Hausdorff and $\mathcal P$ is a partition of $X$. Define the map $\pi:X \to \mathcal P$ taking each point to the unique partition element containing it. Give $\mathcal P$ the ...
- 11.8k
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Order $\omega$ in a dendrite
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
Here $\...
- 151
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Sections with compact support commutes with tensor over a locally compact Hausdorff space. Why can we reduce to the compact case?
I have a question about the proof of the following result in Kashiwara, Schapira, Sheaves on Manifolds:
Proposition 2.5.12 [Let $X$ be a Hausdorff and locally compact space.] Let $A$ be a ring, and ...
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When compacts generate the Borel $\sigma$-algebra, is $X$ $\sigma$-compact?
Let $X$ be a Hausdorff topological space and let $\mathcal K$ denote the family of compact subsets of $X$. Assume that the Borel $\sigma$-algebra $\mathcal B(X)$ is generated by compact sets, i.e.
$$
\...
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Heine-Borel Theorem in Complete Riemannian Manifolds [closed]
Let $(M,g)$ be a complete Riemannian manifold and let $d_g:M^2\to\mathbb R$ be the length-minimizing metric induced by $g$.
Does it necessarily hold that the compact subsets of $M$ are closed and $d_g$...
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When do linear constraints form a compact? [closed]
Given a list of constraints
$$
F_i = \{ x\in\mathbb{R}^n\mid L_i(x) \leq c_i \}
$$
where $L_i\in L(\mathbb{R}^n,\mathbb{R})$ and $c_i\in\mathbb{R}$ for every $i\in\{ 0,\dots,m \}$ with $m \geq n$, ...
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Uniform Compactness Radius for Locally Compact Metric Spaces
I was reading a nice paper, but right at the beginning of one of the main results they claim:
$X$ is a locally compact metric space, so it has an equivalent metric such that every closed ball of ...
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Proving every open cover having finite subcover entails sequential compactness
I am trying to prove that if it is true for a set $K$ that every open cover of it has some finite subcovering, then $K$ is sequentially compact and am looking to verify steps that I'm unsure about in ...
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Exercise on Alexandroff compactification
Let $X,Y$ be Hausdorff and locally compact topological spaces, with $Y$ not compact, and let $\hat{Y}$ be the Alexandroff compactification of $Y$. Let $A\subset X$ be an open, relatively compact ...
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Attribution for Metrization Theorem
Let $f:X \rightarrow Y$ be a continuous surjection with $X$ a compact metric space and $Y$ a Hausdorff space. Then $Y$ is metrizable.
Does this theorem have a name? Who first proved it? Just ...
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Finite and non-compact (non-Hausdorff) cell complex?
$\DeclareMathOperator{\int}{int}$Let $(X,(e_\alpha)_{\alpha \in I})$ be a finite cell complex, i.e. $I$ finite. If $X$ is Hausdorff then the closures of the cells $\bar e_\alpha$ are compact hence $X$ ...
- 7,056
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Equivalent conditions for a metric space to be sequentially compact
I am trying to prove the following theorem
Theorem - Let $\left( X,d \right)$ be a metric space, the following are equivalent:
$X$ is sequentially compact.
Every countable open cover $\left\{ U_{n} \...
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1
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127
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Oscillation of a function on a compact set with a positive bound
Given a function $f:E\subset\mathbb R^n\longrightarrow\mathbb R$ and $x\in E$. The function
$$\omega(f,E):=\sup\{|f(x)-f(y)|:x,y\in E\}$$
is called the oscillation of $f$ on $E$, and
$$\omega(f,x):=\...
- 559
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2
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135
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Every compact metric space is complete - without any a priory knowledge of compactness [duplicate]
I am currently revising my Topology notes, and today I have started looking into compact metric spaces. I am trying to prove explicitly the following:
Theorem - Let $\left( X, d\right)$ be a compact ...
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A closed subset is compact?
Let's suppose $F$ is a closed subset of $E$ or :$$\Leftrightarrow \forall (x_n) \subset F \;,x_n\rightarrow x \in E \implies x \in F$$
A subset of E is compact $$\Leftrightarrow \forall (x_n) \subset ...
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99
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The relationship of compactness between two comparable topologies [closed]
Let $\tau$ and $\tau^{\prime}$ be two topologies on $X$, and $\tau\subset\tau^{\prime}$. I know that if $(X,\tau^{\prime})$ is compact , then $(X,\tau)$ is compact. The proof is also quite ...
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Proof check for compactness of a set of rationals
I was wondering if someone could check this proof.
Consider all rationals p such that $2 < p^2 < 3$ and denote this set E.
Show that the set is not compact in $\mathbb{Q}$.
Let’s just assume I ...
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about locally finite measures and Hausdorff spaces.
In Wikipedia , the following definition is given:
Let $X$ be a locally compact Hausdorff space, and let $\mathfrak{B}(X)$ be the smallest $\sigma$-algebra that contains the open sets of $X$; this is ...
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Prove that all compact subset are closed in a space that all bijections from a compact space are homeomorphisms.
Let $X$ be a compact space such that for any continuous map $f\colon C\rightarrow X$ with $C$ compact space, if $f$ is a continuous bijection then $f$ is an homeomorphisms.
Prove that all compact ...
- 2,064
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1
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Countable sum theorem in dimension theory due to Munkres [duplicate]
I have struggles with an exercise in Topology by Munkres:
Assume $X$ is a Hausdorff, 2nd countable, locally compact space, with the property that each point of $X$ has a neighborhood with closure ...
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3
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1
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74
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Uniform convergence of a functional.
Let $\mathcal{K} \subset L^1{([0,1];\mathbb{R})}$ be compact and define the following sequence $(f_n)_{n \in \mathbb{N}}$ via $f_n \colon \mathcal{K} \to \mathbb{R}$
$$
f_n(u)=\int_0^{1/n} |u(s)| \, \...
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Approximate Equicontinuity and Kolmogorov compactness
Suppose $\{u^n\}$ is a sequence of functions uniformly bounded in $L^1(\mathbb{R})$, and instead of true equicontinuity in the Frechet--Kolmogorov--Riesz compactness sense, we have only an "...
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Does the adjoint of a compact operator map a weak* convergent net to a norm convergent net?
Continuing from the following discussion:
Does the adjoint of a compact operator maps a weak* convergence sequence to norm convergence?
Let $X, Y$ be two Banach spaces and $T:X\to Y$ be a compact ...
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131
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Validity of a potential characterisation of the closed convex hull of a compact set in a Banach space
I am interested in whether the closed convex hull of a compact subset of a Banach space can be characterised in a precise manner described in my question below.
Let $X$ be a Banach space. Given a ...
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300
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How to prove that the distance between a compact and a closed set that are disjoint is positive? [closed]
Problem: Suppose $K$ and $F$ are disjoint sets in a metric space $X$, $K$ is compact and $F$ is closed. Show that there exists a $\delta >0$ such that $d(x,y) > \delta$, for $x \in K$ and $y \in ...
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Continuity of tangent vectors to normal geodesics in Cartan–Hadamard manifolds
Let $M$
be a $3$--dimensional Cartan--Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S\subset M$
is a $C^{1,1}$
surface which encloses a domain $E$.
Let $D_0$...
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A map defined on a compact domain is continuous if and only if its graph is compact. [duplicate]
Problem: If $E$ is a compact subset of a metric space $X$, and $f$ is a map defined on $E$ to a metric space $Y$, then prove that the graph of $f$, denoted by $G(f)$ and defined as the set
$$\{(x,f(x))...
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How to ensure a set of functions in $C[0,1]$ is compact in the sup norm?
Consider the set $C[0,1]$ of continuous functions $f\colon [0,1]\rightarrow \mathbb{R}$, endowed with the topology induced by the norm
$$|f|_\infty = \max_{x\in [0,1]} |f(x)|.$$
Arzelà–Ascoli theorem ...
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2
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If $X$ and $Y$ are compact Hausdorff spaces and $f \colon X \longrightarrow Y$ has closed graph, then $f$ is continuous
Let $X$ and $Y$ be topological spaces such that both $X$ and $Y$ are both compact and Hausdorff, and let $f \colon X \longrightarrow Y$ be a mapping such that the subset
$$
\big\{(x, y)\in X \times Y \...
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Is every bounded subset of $X^*$ relatively weak$^*$-compact?
Let $X$ be a Banach space and let $Y \subseteq X^*$.
I came across the following proposition:
Proposition. If $Y$ is bounded in $X^*$, then $Y$ is relatively compact in $X^*$ with respect to the weak*...
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Do co-finite neighborhoods characterize spaces with all subsets compact?
In e.g How is every subset of the set of reals with the finite complement topology compact? it's shown that a space with the co-finite topology, or more generally any topology where neighborhoods are ...
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Density of functions with compact support over continuous functions vanishing at infinity
Let $\mathrm{V}, \mathrm{W}$ be two normed spaces. Consider the following sets of functions $\mathrm{V} \to \mathrm{W}$. First, $\mathscr{K}$ is the set of continuous functions with compact support ...
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How can a vector space be compact?
Problem 9, Section 4.13 of Kreyszig's 1978 edition of Introductory Functional Analysis with Applications states:
If $T: X \to Y$ is a closed linear operator, where $X$ and $Y$ are normed spaces and $...
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Open dense subset of a compact lie group has full measure
For a given compact (lie) group there exists a unique haar propability measure $\mu_G$.
Is it true that any open dense subset $U\subseteq G$ has full measure, i.e. $\mu_G(U) = 1$?
I know the statement ...
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Quotients of compact metric spaces whose projection maps are local homeomorphisms
Relevant Definitions
A map of topological spaces $f: X \to Y$ is said to be a local homeomorphism if for each $x \in X$ there exists some open set $U \subseteq X$ with $x \in U$ such that $f(U) \...
- 4,970
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Relation between free arcs and branch points in continua
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
...
- 151
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Countably compact space where every compact set is finite
Is there an example of an infinite, countably compact topological space $X$ with the property that every compact subspace $K\subseteq X$ is necessarily finite? In some sense, this would be the ...
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100
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Free arcs in continua
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
...
- 151
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2
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Generalized Heine-Borel Theorem - Understanding proof
From T. B. Singh's Introduction to Topology book:
Theorem 5.1.16 (Generalized Heine-Borel Theorem) A closed and bounded subset $A$ of the Euclidean space $\mathbb R^n$ is compact, and conversely.
...
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1
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Strongly collectionwise normal spaces with $G_\delta$ diagonal are submetrizable
There is a classic result by Chaber which says that a countably compact Hausdorff space with $G_\delta$ diagonal is metrizable. On Dan Ma's blog one can find that a countably compact space with $G_\...
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Continuity of the Infinite-Dimensional Representation of $SO(3)$ in $L^2(\mathbb{R}^3)$
I'm new and not a mathematician, so I apologize for the question, which may seem trivial to you. However, I would like to demonstrate, without invoking concepts that, unfortunately, I haven't studied ...
- 11
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1
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A result about dendrites.
before I begin, I would like to provide some definitions and a theorem.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
...
- 151
2
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1
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170
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Finite product of compact topological spaces is compact - understanding notation in proof
Let $X$, $Y$ be compact topological spaces, then so is $X\times Y$ (equipped with the product topology). In his book Introduction to Topology, T. B. Singh starts as follows (cf. Theorem 5.1.15):
...
- 1,107
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1
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124
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Covering Properties of a Topological space
Let $ (X, \tau) $ be a topological space.
$ (X, \tau) $ is said to be a compact space if every open cover of a space $ X $ has finite subcover;
$ (X, \tau) $ is said to be a countably compact space ...
- 11
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0
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138
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Direct proof of [second-countable $\wedge$ sequentially compact $\to$ compact]
Let's talk about topological spaces and their different notions of compactness.
It is known that any second-countable and sequentially compact space is compact. I can only find indirect proofs of it, ...
- 1,189
2
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1
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98
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Help with my proof of the Krein-Smulian theorem for convex hulls of weakly compact subsets of real Banach spaces
Let $V$ be a real Banach space with squared norm $|| \cdot ||^2 : V \rightarrow \mathbb{R}$. From the Hahn-Banach theorem we know that
$$\forall v \in V, || v ||^2 = \text{sup} \left\{ \phi(v)^2 \ \...
user1654888
1
vote
1
answer
56
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Integral condition for tightness of probability measures.
In the book Gradient flow by Ambrosio, Gigli and Savare. There is a useful condition (Remark $5.1.5$, an integral condition for tightness). More precise, to check a set $K\subseteq \mathbb{P}(X)$ is ...
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