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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.

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Compactness of the set of finite Borel measures

Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n \in \mathbb N$. Let $\mathcal M(X)$ denote the space of all finite Borel measures on $X$. Is $\mathcal M(X)$ compact under some commonly ...
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1answer
41 views

The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology.

The book "Invitation to Algebraic Geometry" says the following: The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology. Why is this this the case? This is thing that is asked ...
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1answer
27 views

compactness proof in Munkres

In Munkres, he proves that every closed subspace Y of a compact space X is compact. In the proof, he adjoins the open set X - Y to an open cover of Y, and then he goes on to prove Y is compact. Why ...
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1answer
26 views

Example of Non-Compact Closed Set contained in Open Set With Special Property

I am looking for an example of an open set $A$ in a metric space $X$ and a closed, non-compact subset $B$ of $A$ such that there is no $\delta > 0$ s.t. $\{x: \textrm{dist}(x,B) < \delta\} \...
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1answer
39 views

If $f\in C_0$ and $g$ is continuous, can we show that $f\circ g\in C_0$?

Let $E$ be a locally compact Hausdorff space and $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\}.$$ Let $f\in C_0(E)$, $E'$ be another ...
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0answers
37 views

Hausdorff, locally connected and locally compact space reference.

I would like to find a reference to the following proposition: Let $X$ a Hausdorff, locally compact, locally connected, connected space and $K \subseteq X$ a compact subset. Then, there exists a ...
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1answer
51 views

Is Hausdorff condition necessary to solve this problem?

Suppose that $X$ is a compact Hausdorff space and let $\mathcal{F}\subseteq\mathcal{P}(X)$ be a family of closed sets in $X$ whit the FIP property. Let $U\subseteq X$ be an open set such that $\bigcap\...
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0answers
23 views

Uniform Continuity on Compact Set with twist

I have been asked to prove the following: Have $g:[a,b]\rightarrow \mathbb{R}$ be continuous on $K = [a,b] \setminus \cup^\infty_{n=1} (\alpha_n,\beta_n)$. Then, for any $\epsilon > 0$, there is a ...
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0answers
26 views

Show that any subspace of a compact space can be covered with one open subspace.

Here's the problem I'm dealing with: Let $(X,d)$ be a compact metric space and let $(U_{\lambda})_{\lambda \in \Lambda}$ be an open cover of $X$. Show that there exist $\delta >0$ such that for ...
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1answer
34 views

Real Analysis - proving compactness

Let $K⊆R^n$ be a set such that every infinite subset of $K$ has a limit point in $K$. How can we show that K is closed and bounded?
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Frechet Derivative and Convergence of Functionals

Let $\Omega\subset\mathbb{R}$ be a bounded interval, $\{u_{n}(t)\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ and define $J[u_{n}(t)] = \frac{1}{2}||u_{n}(t)||_{H_{0}^{1}(\Omega)}^{...
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1answer
46 views

Property of compact set K contained in an open set G

I'm trying to show that if $G$ is an open set and $K$ is a compact set with $K \subset G$, show that there is a $\delta > 0$ such that $\{x: \textrm{dist}(x, K) < \delta \} \subset G$. My ...
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1answer
30 views

Equivalent conditions for relative compactness in the Skorohod space

Let $(E,d)$ be a complete separable metric space and $\mathcal X$ be a family of càdlàg functions $E\to[0,\infty)$. Consider the following claim: For all $t\in[0,\infty)\cap\mathbb Q$, there is a ...
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3answers
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Is there a version of the Arzelà–Ascoli theorem capturing $C([0,\infty))$?

I only know the Arzelà–Ascoli theorem for continuous functions on a compact topological space. However, in the context of characterizing weak convergence of probability measures on $C([0,\infty))$, I'...
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1answer
14 views

Query regarding a compact set in a open set $\Omega$ in $\Bbb{C}$

Suppose $\Omega$ is a bounded open set in $\Bbb{C}$ For any $\delta>0$ let us define a new set $\Omega_\delta=\{z\in\Omega|\overline{D_\delta(z)}\subset\Omega\}$. Is $\Omega_\delta$ compact? ...
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1answer
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Compact Hausdorff space is metrizable if there countable separating continuous functions

Proposition: Let $X$ be a compact Hausdorff space. Suppose there are countable real valued continuous functions $\{f_n\}_{n \in \mathbb{Z}_+}$ separating $X$ i.e. for all $x, y \in X$ with $x \neq y$, ...
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1answer
64 views

Conjecture: Is it true that the limiting diameter of a (strictly !) nested family of compact sets must be 0 ??

Consider a family of compact subsets of $\mathbb{R}^n, C_1 \supset C_2 \supset C_3 \ldots$. Also, and this is the important bit, 1) $C_j$ has empty interior for all $j \in \mathbb{N}$ 2) The ...
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1answer
38 views

Real Analysis Question, continuous functions over R with period 2π [closed]

Before getting into the question, just to let you guys know that I have a final tomorrow and this was on the past test but I'm lost as to how to proceed on this, so any help will be great! Edit - I ...
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0answers
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Prob. 2, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: Local compactness of a product of locally compact spaces

Here is Prob. 2, Sec. 29, in the book Topology by James R. Munkres, 2nd edition: Let $\left\{ \ X_\alpha \ \right\}$ be an indexed family of nonempty spaces. (a) Show that if $\prod X_\alpha$ ...
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1answer
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Prob. 1, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathbb{Q}$ of $\mathbb{R}$ is not locally compact

Here is Prob. 1, Sec. 29, in the book Topology by James R. Munkres, 2nd edition: Show that the rationals $\mathbb{Q}$ are not locally compact. My Attempt: Here the topology on the set $\mathbb{...
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3answers
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How $A=\{x_n:n\in \mathbb N\}$ is an infinite set?Can you explain the proof?

Can you please explain the underlined arguments? How does it deduce that $A$ is infinite? I understood that when $y\in X$. There is an $U\in \mathscr O$ such that $x\in U$(since $\mathscr O$ is an ...
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1answer
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A lemma used to prove the classification of closed 1-manifolds

In our lecture notes, we have the following lemma which is afterwards used to prove the classification of closed 1-manifolds (Compact Hausdorff spaces every point of which has a neighbourhood ...
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1answer
30 views

Relation between strong convergence in $L^{p}$ and weak convergence in $H_{0}^{1}(\Omega)$

Let $\{u_{n}\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ for a bounded interval $\Omega \subset \mathbb{R}$. By weak compactness of Hilbert Space, we can extract a subsequence of $...
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0answers
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Meaning of Compactness

Let $\Omega \subset\mathbb{R}$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = ...
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2answers
19 views

Compact sets through topologies over a set

I have to show that a set is compact on the Sorgenfrey line. I can prove that it is indeed compact on the usual topology over the real line and my idea is to say that, since the Sorgenfrey topology is ...
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1answer
30 views

intersection of decreasing compact sets is again compact

I am following a real analysis course and I would like to discuss some definitions and results about metric spaces. First of all, the definition of compact set. I the course we defined it as follows: ...
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1answer
18 views

Regarding sequence of positive Harmonic functions

Let $\{U_n\}_{n\geq 1}$ be a sequence of positive harmonic functions on a domain $\Omega$ and let $z_0\in \Omega$. Suppose that $\lim_{n\longrightarrow \infty}U_n(z_0)=\infty$. How does one show that $...
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2answers
202 views

Showing the closure of a compact subset need not be compact

Could someone tell me if my line of reasoning is correct here: Say we have the topological space $(\mathbb{N}, T)$ comprising of the empty set together with all subsets of $\Bbb N$ that contain the ...
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2answers
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$d(x,E^c) > \frac{1}{j}$ is compact if $E$ is open.

It is claimed in an analysis text that Let $E \subseteq \Bbb R^n$ be an open set. Then $$K_j := \{ x \, :\, d(x, E^c) \ge 1/j \}$$ is a compact set. How does one see this? I guess it is ...
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1answer
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Example 4, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: How is the one-point compactification of the real line homeomorphic with the circle?

Here is Theorem 29.1 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a [topological] space. Then $X$ is locally compact Hausdorff if and only if there exists a [topological] space ...
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2answers
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Prove that the one point compactification of $X$ \ $E$ is homeomorphic to $X/ \sim$

Let $E$ be a closed subset of a compact Hausdorff space $X$. Prove that the quotient space obtained from $X$ by identifying $E$ to a point is homeomorphic to the one point compactification of $X $ \ $...
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1answer
37 views

Equivalent Definitions of Compactly Supported Forms in The Vertical Direction

I've come across two definitions of compactly supported forms in the vertical direction and I'm trying to show they are equivalent. For the setup, let $\pi:E \to M$ be a vector bundle of smooth ...
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1answer
36 views

Independence of the uniform metric from the compatible metric for the codomain

Let $X$ be a compact metrizable space and $Y$ a metrizable space. Denote by $C(X,Y)$ the space of continuous functions from $X$ to $Y$ with the topology induced by the uniform metric $$d_u(f,g)=sup_{x\...
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1answer
62 views

What is application of following exercise?

I had done following excercise. Consider the function $f:X\to Y$ where Y is compact Hausdorff space. Then $f$ is continuous if and only if the graph of $f$, $$G_f=\{(x,f(x)) \mid x\in X\},$$ is ...
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1answer
57 views

Examples of sequential compact but not compact spaces that do not use ordinals.

I think the title is self explanatory, I'm using Munkres' Second Edition text for Point Set Topology and I can't figure out if such examples are possible.
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1answer
33 views

Noncompactness of $L^{p}$ space in $[-1,1]$ interval

I want to show that for $1\leq p < \infty$, $L^{p}([-1,1])$ is not compact. I have found an example here ( Unit sphere in $L^p([0,1])$ is not compact. )by constructing the following sequence \...
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1answer
29 views

Relatively compact and compact set exercise

I have che following exercise and some dubts: Let $M>0$ and $\mathcal{F}=\{ f\in C^{1}([a,b]) \, | \, \| f \|_{C^{1}} \leq M\}$. Prove that $\mathcal{F}$ is relatively compact in $(C^{0}([a,b]), \|...
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Is the set $\overline{\text{conv}}^{w^*} C$ weakly* compact?

Exercise : Let $X$ be a Banach space and $C \subseteq X^*$ be $w^*-$compact. Is the set $\overline{\text{conv}}^{w^*} C$ $w^*-$compact ? Thoughts : I (think) that I know that $w^*-$compact sets ...
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2answers
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X is sequentially compact $\implies $ then Lebesgue lemma hold for X where X is metric space

X is sequentially compact $\implies $ then Lebesgue lemma hold for X where X is metric space Lebesgue lemma:X is said to satisfy Lebesgue number lemma it for every open cover of X there in $\...
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1answer
37 views

Inverse image of singleton sets under continuous map between compact Hausdorff topological spaces

Let $X,Y$ be non-empty compact and Hausdorff topological spaces and $f:X \to Y$ be a continuous map. Take an element $y \in Y$. Question: Is $f^{-1}(\{y\})$ closed in $X$? Approaches and Ideas (...
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1answer
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Show that $S_R := \{x \in \mathbb{R}^n : \|DL_x\| = R\}$ is compact

Let $L : \mathbb{R}^n \to \mathbb{R}$ be a convex smooth function such that its derivative $DL : \mathbb{R}^n \to (\mathbb{R}^n)^*$ is diffeomorphism. For each $R > 0$ define $$S_R := \{x \in \...
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2answers
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Compactness of a set : $A=\left\lbrace 2019,1010,\ldots,(n+2018)/n,\ldots \right\rbrace$ and $B=A \cup \left\lbrace 1 \right\rbrace$ [duplicate]

How can I show directly (that is, not using the compactness criterion in $\mathbb{R}$) that every open cover $G$ of $B$ admits a finite subcover? I know that $A$ is not compact and I am aware that I ...
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2answers
64 views

Is a compact union of compact spaces still compact?

I'm wondering if the following statement is true : Let $ K $ be a compact space, $ X $ a topological space and $ \forall x $ in $K$, $L(x)$ a compact subspace of $ X $. Then $ \cup_{x \in K} L(x) $ ...
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1answer
25 views

Regularity of measure on locally compact separable metric space

Proposition: Let $X$ be a locally compact separable metric space, and $\mathscr{B}$ be Borel $\sigma$ -field of $X$. Then radon measure $m$ on $(X,\mathscr{B})$ is regular. Here, we ...
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1answer
55 views

A quotient space by group action is manifold $???$

The aim of this question is to construct a exmaple of quotient manifold. First, I set notations and difnitions. Let $\mathbb{R} , \mathbb{C}$ be real and complex numbers with usual topology. We ...
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0answers
19 views

Find a Function with compact support

Show that there exists $\phi \in C_{c}^{\infty}(R^n)$ such that $0\leq \phi(x)\leq1 $ for all $x$ and $\phi(x)=1$ if $|x|\leq 1$ and $\phi(x)=0$ if $|x|\geq2$. I want to use the function $\eta(t)=e^{-...
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1answer
23 views

Let X be sequentially compact. Then X is also countably compact.

A subset $A $ of a topological space $X$ is sequentially compact iff every sequence in $A $ has a convergent subsequence. Countably compact means every countable open cover has a finite subcover. I ...
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1answer
22 views

Compactness implies countably compactness

Let X be compact. Then X is countably compact. My thinking is like this: Let X be a topological space. Since compact, then every open cover hava a finite subcover. Hence it is true for countable open ...
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1answer
51 views

Prove that Sequentially Compact Metric Spaces are Lindelöf without the Axiom of Choice.

A proof can be found here, but it seems that it uses AC. I would like to know if there is a proof for this fact without AC. I came up with this question after seeing a proof of sequential compactness ...
3
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1answer
32 views

uniformly continuous imply bounded

Proposition: Let $(X,d)$ be compact metric space, and $Y$ be Borel subset of $X$. Suppose $A$ is homeomorophic to $Y$. Then, uniformly continuous function $f:A \to \mathbb{R}$ is bounded function. I ...