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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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27 views

Using Alexander lemma to prove that if $X$ is continuum then the hyperspace $2^X$ is compact

So far in the books I've read all proofs involving Alexander Lemma to prove Tychonoff's theorem or in general any compact space they use Zorn Lemma argument. So I was wondering if is there any ...
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2answers
39 views

Let $f:K \rightarrow N$ be a continuous function from a compact $K$. Show that $f$ is uniformly continuous

I'm having trouble finishing this. One approach that I made is this: Let $\epsilon > 0$. Then, since $f$ is continuous, for every $x \in K$ exists $\delta_x > 0$ such that $d(x, x')<\delta_x ...
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1answer
48 views

Prove that this operator is compact

Let $E$ be the real Banach space of all real and continuous $\omega $-periodic functions defined on $\mathbb{R}$ with the norm $$\max_{0\leq t\leq\omega}\left | x(t) \right | \:,\:\forall x\in E$$ ...
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0answers
34 views

Is there a name for this notion of “radius of compactness” in a metric space?

I was proving some result about Riemannian manifolds that led me to introduce the following definition: Let $M$ be a metric space and $x \in M$. Define the "radius of compactness" $RC(x)$ to be the ...
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1answer
45 views

Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
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1answer
38 views

On the dimension of rationals over real numbers

The following two are from Introductory Functional Analysis by E Kreyszig: 2.5-2 Lemma . A [sequentially] compact subset M of a metric space is closed and bounded. The converse of this ...
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3answers
222 views

Covering a compact set with balls whose centers do not belong to other balls.

Let $K\subset \Bbb R^n$ be a compact set such that each $x\in K$ is associated with a positive number $r_x>0$. Claim: $K$ can be covered by a family of balls $$ \mathcal B = \{ B(x_i,r_i) : i=...
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1answer
30 views

Is every metric space compact? [duplicate]

I am referring to Rudin's definition 2.32 of compactness here: A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Obviously X is a subset of X ...
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1answer
30 views

In a locally compact 2nd-countable Hausdorff space $E$ there is a sequence of compact subsets $K_n$ with $K_{n-1}⊆\overset∘{K_n}$ and $\bigcup_nK_n=E$

Let $(E,\tau)$ be a locally compact second-countable Hausdorff space. I want to show that there is a $(K_n)_{n\in\mathbb N_0}\subseteq E$ such that $K_n$ is compact and $K_{n-1}\subseteq\overset{\circ}...
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1answer
32 views

Visual representation of difference between closed, bounded and compact sets

I have trouble grasping the difference between bounded, closed and compact sets. As a picture is worth a thousand words (especially for a person with a light math background), I would like to get a ...
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1answer
179 views

Counterexample for “continuous image of closed and bounded is closed and bounded” (in normed spaces).

It's well known that: If $X$ is a finite-dimensional normed space, $C$ is a closed and bounded subset of $X$ and $f:C\subset X\to X$ is continuous, then $f(C)$ is closed and bounded. If $X$ is any ...
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1answer
30 views

Does the given set is compact in $\mathbb{R^{2}}$?

Check whether given set on $\mathbb{R^{2}}$ is compact or not ? $ \left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\sin\left(\frac{1}{x}\right)\right\}\bigcap\left\{(x,y) \in \mathbb{R^{2}}\,\...
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2answers
45 views

Two disjoint closed sets $A,B$ with $B$ compact, show $d(A,B) > 0$ Verify my proof

Two disjoint closed sets $E,F$ with $E$ compact, show $d(E,F) > 0$. So, with compactness we get a few things, that every sequence has a convergent subsequence and we can use the extreme value ...
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1answer
36 views

Prove that this space is compact/not compact by using Arzelà–Ascoli theorem

I need to tell if this space/set is compact in $C[0,1]$ $x_n(t) = t^n, n ∈ N$ Following Arzelà–Ascoli theorem, the set is compact when it has Uniform boundedness and Equicontinuity, is it correct? ...
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1answer
22 views

A $T_{3 \frac{1}{2}}$ Lindelöf space $X$ is normally placed in $\beta X$

A subset $X \subset Y$ is normally placed in $Y$ if whenever $X \subset U$ for $U$ open in $Y$ there are $F_n$ closed in $Y$ such that $X \subset \bigcup_{n \in \mathbb{N}}F_n \subset U$. Let $X$ ...
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2answers
28 views

Sequence of points in a nested sequence of sets converges to the point in the limit of the sequence of sets.

Let $B_n$ be a decreasing sequence of compact subsets of a metric space convergent to compact set $B$. That is $B_{n+1}\subseteq B_n$ for all $n$ and $\bigcap\limits_{n\geq 1}B_n = B$. Let $b_n\in B_n$...
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1answer
36 views

Example 1.44 In Rudin's Functional Analysis (“Exhaustion by Compact Sets”)

In example 1.44, Rudin introduces the space $C(\Omega)$, where $\Omega \subseteq \Bbb{R}^k$ is open. He claims that $\Omega$ can be written as a countable union of compact sets $\{K_n\}_{n \in \Bbb{N}}...
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3answers
351 views

Compactness in normed vector spaces.

Let $(X,\Vert\cdot\Vert)$ be a normed $\mathbb{K}$-vector space and $A \subset X$ be closed and bounded. My problem is how to determine whether $A$ is compact? I know that a compact subset is always ...
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1answer
31 views

Locally compact limits of countably many finite $T_0$-spaces

Consider any diagram of finite $T_0$-spaces where the spaces involved are at most countably many. So we have $X_0, X_1, \ldots, X_n, \ldots$ each of which is $T_0$ and finite and some continuous maps ...
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1answer
35 views

Counter-example Aubin-Lions p=1

I wanted to show, that $p>1$ in Aubin-Lions Lemma is necessary. I hoped there already is an example for $X=Y=Z=\mathbb{R}$ so that the embedding \begin{align} \{u \in L^\infty(0,T);\mathbb{R}) \...
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1answer
27 views

Showing $[0,1]$ is compact w.r.t to cofinite topology [duplicate]

Let $\tau:=\{A\subseteq \mathbb R: \mathbb R\setminus A \operatorname{finite}\}$ Show that $[0,1]$ is compact. Let $(A_{n})_{n}$ be an open cover of $[0,1]$ I had two ideas: $1.$ Since $[0,1]\...
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1answer
18 views

Compact space homotopy equivalent to a CW complex

Assume that $X$ is a compact Hausdorff space homotopy equivalent to some CW complex. Does it follow that it is homotopy equivalent to a compact CW complex?
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1answer
105 views

Closed maps with non-compact fibers

A geometric example of a non-closed map with compact fibers is obtained by taking a finite covering map and removing a point upstairs. This fragmentation of the fiber destroys closedness but retains ...
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1answer
21 views

I wondering how can I build a homeomorphism between two squares and a ellipsoid.

For example, if we have X equal the union on the squares $[0,1] \mbox{x}[0,1]\mbox{x}{0}$ and $[0,1] \mbox{x}[0,1]\mbox{x}{1}$ and $Y= (x^2)/4 + (y^2)/4 + (z^2)/1 = 1$. How can I construct a ...
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1answer
18 views

Region between two continuous curves is compact

Suppose that there are two continuous functions $\phi_1,\phi_2:A\subseteq\mathbb{R}^n\to\mathbb{R}$ and the set $$ D=\left\lbrace x\in A:\phi_1(x)\leq\phi_2(x)\right\rbrace $$ is compact and Jordan ...
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3answers
57 views

Maximal Counterexample to Compactness

The question is: Let $X$ be not compact. Prove that there exists an open cover $A$ such that $A$ does not have a finite subcover and any open cover $A \subsetneq B$ has a finite subcover. I am ...
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56 views

$f:X\rightarrow Y$, with $Y$ compact, is continuous $\Rightarrow$ its graph is closed [duplicate]

Let $X,Y$ topological spaces, $f:X\rightarrow Y$ a function, with $Y$ compact. I need to show that, if $f$ is continuous, so $$G_{f}:=\{(x,f(x)):x\in X\}$$ is closed in $X\times Y$. I saw that ...
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2answers
46 views

Railway metric, help showing that the space is bounded but not compact

I am studying metric spaces and I came across this exercice that I was able to solve only partially. Any help and hints will be appreciated. Let $X= \{ x \in \mathbb{R}^2 | \mid \mid x \mid \mid \...
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1answer
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Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact. [duplicate]

Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact. My attempt:- We know that $(X,\mathscr T)$ is metrizable. So ...
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0answers
16 views

Brunn-Minkowski Inequality

Let $K$ be a convex compact set in $\mathbb{R}^n$, and let $$F(t)=m_{n-1}(K\cap\{x\in\mathbb{R}^n:\,x_1=t\}).$$ Using Brunn-Minkowski inequality in its third formulation, prove that the function $\log ...
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1answer
12 views

Equivalence of total boundedness and relative compactness in Polish spaces.

Let $E$ be a Polish space. A set $A\subset E$ is totally bounded if and only if $A$ is relatively compact. First we suppose $A$ is totally bounded. Since $\overline A$ is closed and $E$ is complete, $...
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0answers
42 views

Behavior of points and compact subsets of Hausdorff spaces

It is quite straightforwad to see that many prpoperties are shared by points and compact subspaces of Hausdorff topologies, for example in terms of separation properties. I was wondering if there is ...
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2answers
36 views

A question about compact support

Support is defined as the closure of $\{x:f(x)\neq 0\}$. Now consider $f(x)=\frac{1}{x}\chi_{(0,1)}$. So the support is $[0,1]$ which is compact. However, any continuous function with compact support ...
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1answer
36 views

First-order Peano arithmetic and (the lack of) implicit definition of addition

I'm trying to show, through the existence of non-standard models of arithmetic, that the first-order Peano axioms (without those of multiplication) don't implicitly define addition in the sense of ...
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0answers
36 views

Show that a closed graph and compact codomain implies continuity. [duplicate]

Let $$f:X\rightarrow Y,$$ where X and Y are metric spaces and Y is compact. If f has a closed graph $$G_f=\{(x,f(x)):x\in X\},$$ then f is continuous. My first instinct is to use the fact that if ...
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1answer
33 views

Sequentially compact metric space is totally bounded.

I want to prove this: " If for any sequence $(x_n)$ from a metric space $(E,d)$ we can extract a convergent subsequence then for any $r>0$, we can cover $E$ by a finite number of open balls of ...
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1answer
20 views

Sequential definition of compactness: is the accumulation w.r.t. the sequence or the underlying set?

In Lang's Complex Analysis, page 21, he gives the following definition of compactness: We define a set of complex numbers to be compact if every sequence of elements of S has an accumulation ...
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1answer
46 views

Does every topological space have an open cover?

Is it guaranteed that any topological space would always have an open cover? I think it should, but I wanted to check why. I feel like it's maybe related to the base of a topology? I know the base ...
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1answer
28 views

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
47 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
28 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
51 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
54 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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0answers
18 views

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
47 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 ...