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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Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
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1answer
49 views

A question about closed (but not necessarily compact) connected subsets of Euclidean spaces.

Is the following statement true?...... If $C$ is a non-degenerate closed and connected subset of the Euclidean plane $\mathbb R ^2$ and $p$ is any point of $C$, then there exists a connected ...
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63 views

Sequential compactness implies compactness: what is wrong with this argument?

Definitions A filter is a poset $(I,\leq)$ such that for any $\alpha,\beta\in I$ there is $\gamma\in I$ such that $\gamma\geq\beta,\gamma\geq\alpha$. A net in a set $X$ is a function from a poset to ...
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1answer
51 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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0answers
24 views

About accumulation point in compact metric space

Let $(X, d)$ be a compact metric space, and $\{x_n\}_{n\in N}$, $\{y_{n,m}\}_{m,n\in N}$ be subsets of $X$. Question: Is there a subsequence $\{n_k\}$ such that $x_{n_k}\rightarrow x$ and ...
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2answers
27 views

E is infinite subset of compact set, then is E' also a subset?

Here's a theorem in Rudin's Principles of Mathematical Analysis. 2.37 Theorem: If E is an infinite subset of a compact set K, then E has a limit point in K. Proof: If no point of K were a limit ...
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1answer
26 views

Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ ...
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1answer
30 views

Normalized measure over compact metric spaces

Consider the following definitions. Let $M = (V,T,d)$ be a compact metric space with finite diameter $$D = D(M) = \max d(x,y), ( x, y \in M)$$ and a finite normalized measure $\mu$$M$(.), ...
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1answer
25 views

Why is the the following statement not equivalent to compactness?

Well it comes down to word-play again. I'm confused to the core of my bones as to why the following isn't equivalent to saying that a space is compact Every open cover is finite. A compact set ...
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70 views

How to define $[-\infty, \infty]$ or $[0, \infty]$?

I am familiar with basic undergraduate topology. For example, I know the process of one point compactification of a non-compact topological space, and how it applies to, say, $\mathbb R^2$. My ...
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1answer
29 views

Discuss about compactness of these sets

My question is: How can I see if (in $\mathcal H=\mathcal l (\mathbb{N} )$ $B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$ ,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$ are compacts or ...
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1answer
28 views

Proof of being a compact set [closed]

I'm trying to solve this problem but I'm really stuck and it would be nice if someone can explain me proof or any hint for this problem. Let $X \subset\mathbb R^N$ be a nonempty compact set, and $f: ...
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3answers
90 views

Why is $ \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ not compact?

$ S = \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ is obviously bounded and infinite. It also looks totally disconnected to me (it is not and does not contain as its subset an interval with more ...
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1answer
37 views

Does compact set have always content

A set is said to have content iff it's boundary have content zero. So does compact set have always content? I can't really find a way to proof this, but for all examples that I can think of ...
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1answer
26 views

Which of the following are compact I need Hint…

Which of the following are compact? $\{(x,y) \in \mathbb{R}^2 :(x-1)^2+(y-2)^2=9\} \cup \{(x,y) \in \mathbb{R}^2: y=3\}$. 2.$\{(\frac{1}{m},\frac{1}{n}) \in \mathbb{R}^2:m,n\in \mathbb{Z}-\{0\}\} ...
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0answers
29 views

What is the $\epsilon$ neighborhood of a subset in $\mathbb R^2$ in $\mathbb R^n$

Let X denote the subset $(-1,1) \times 0$ of $\mathbb R^2$ and let U be the open ball B(0,1) in $\mathbb R^2$ which contains X. Show that there is no $\epsilon > 0$ s.t the $\epsilon$-neighborhood ...
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2answers
51 views

How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
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1answer
38 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
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1answer
46 views

Show that a sequence of uniformly bounded continuous functions with Lipschitz condition is pre-compact in the space of bounded continuous functions

I am attempting to solve the following problem: Let the sequence of continuous functions $\{x_{n}(t) \}_{n=1}^{\infty}$, $0 \leq t < \infty$ be uniformly bounded on $t \in [0, \infty)$ and on ...
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1answer
68 views

Why must the countable sets shown in this example be closed?

In the book Introductory Real Analysis (Kolmogorov & Fomin, English Translation by Richard Silverman Pg 95, see below) It says: "the sets $X_n= \{x_n, x_{n+1}, ...\}$ form a centered system of ...
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0answers
25 views

Show that $ \operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\} $ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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2answers
43 views

Question about product of compact spaces being compact

I'm reading through a proof that for $X,Y$ compact, $X\times Y$ compact with the standard topology. The proof begins by saying: consider a specific type of cover $\mathcal{C}$ where each $T\subseteq ...
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9 views

Diagonal argument involving relative compactness of densities

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], ...
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1answer
22 views

How to index compact set problem?

For any α∈I, if Aα is compact set then ∩(α∈I)Aα is compact set. Tomorrow, I will mid-exam, i'm very dizzy. Please, prove that problem.
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17 views

Closed graph of a function

I have difficulties to answer at that question: Let $X$ be a Hausdorff and compact topological space, and let $Y$ be a topological space. Let $f:X→Y$ be such that $G(f) = \{(x,f(x))|x∈X\} $ is a ...
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2answers
26 views

Uniform convergence on compact intervals of R [closed]

Does the sequence $f_n(x)=e^x(1+x/n)$ converge uniformly on compact intervals of R?
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41 views

Show that well-ordering is not a first-order property.

Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup ...
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52 views

If $f$ is defined on $R$ and $f(K)$ is compact whenever $K$ is compact, then is $f$ continuous on $[a,b]$?

If $f$ is defined on $R$ and $f(K)$ is compact whenever $K$ is compact, then is $f$ continuous on $[a,b]$? I know that if $f : K → R$ is continuous and $K \in R$ is compact, then $f(K)$ is compact, ...
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1answer
40 views

Can someone suggest any way to complete (if possible) the following proof of the fact that in a not complete metric space is not compact?

Problem. If $(X,d)$ is a metric space such that it is not complete then prove that $X$ is not compact. My Attempt. Since $(X,d)$ is not complete, there exists a Cauchy sequence ...
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1answer
22 views

Surjectiveness in a compact subset

I am completely at a loss as how to proceed. I can't use differentiability here.The question is Let $K$ be a compact subset of $\mathbb{R}$ and $f:K\rightarrow K$ be a function satisfying the ...
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1answer
51 views

‘Every continuous real-valued function on $X$ achieves a minimum’ is a topological property.

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ takes on a minimum value. I need to show that any topological space that is homeomorphic to $X$ ...
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3answers
18 views

Showing Component of Superlevel Set is Compact

Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x, y) = \frac{x^2}{\left(x^2 + y^2 + 1\right)^3}$. The superlevel set $$D = \left\{(x,y) \in \mathbb{R}^2 \colon f(x,y) \geq ...
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55 views

Can every compact subset of $\Bbb R^n$ be written as a disjoint union of compact subsets, where each of them are path-connected?

I was wondering if every compact subset of $\Bbb R^n$ could be written as a disjoint union of compact subsets, where each of them are path-connected, i.e. : If $X \subset \Bbb R^n$, $n \ge 1$, $X$ is ...
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21 views

Prove compact set and index set

\begin{equation} \forall α \in I\; A_α \text{ is a compact set} \implies \bigcap \{ A_\alpha : \alpha \in I\} \text{ is a compact set} \end{equation} How to prove this problem? I know definition of ...
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1answer
50 views

If $A$ is compact then $f^{-1}(A)$ compact?

Let $f$ be a continuous function. I know that if $A$ is compact then $f(A)$ is compact but is $f^{-1}(A)$ also compact? I believe it is not but how can I prove it by a counter example?
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1answer
17 views

Compact set of closed subset problem

K is subset of complex plane. K is compact set and A is closed subset of K then A is compact set. How to prove this problem?? I know definition of compact set, but i'm not use definition to problem. ...
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18 views

How to prove intersection compact problem? [duplicate]

If A and B are compact set, then A∩B is compact set. How to prove this problem?. I know compact, but not use to this problem...
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1answer
48 views

Is my proof of the fact that the product of two compact metric spaces is compact correct?

Sometimes ago I have posted this question. After sometime of working I think that I have found out a different proof (not "purely topological"). I didn't post it there as an answer because (1) the ...
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1answer
48 views

Topological proof of the compactness of product metric space

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric spaces (see the definition here). Then show that the product metric space $(X\times Y,d_{X\times Y})$ is also compact. Now this can be ...
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1answer
42 views

Prove that $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ is not compact

Let $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ Show that $K$ is not compact
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Is every compact space hereditarily Lindelöf?

All spaces are assumed Hausdorff. We call a topological space compact if every open cover has a finite subcover. We call it Lindelöf if every open cover has a countable subcover, and hereditarily ...
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41 views

Show that a compact metric space $X$ is locally compact

Assume that $X$ is a compact metric space, that is by definition, every sequence in $X$ has a convergent subsequence. Locally compact means that every point in $X$ has a compact neighbourhood. That ...
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3answers
48 views

Tips on determining compact to non-compact sets

I find it hard to quickly determine this. For instance, $D^n/\{0\}=\{x \in \mathbb{R}^n| 0 < ||x|| \leq 1\}$ is non-compact. $S^{n-1}=\{x \in \mathbb{R}^n | ||x||=1\}$ is compact. So, ...
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0answers
26 views

calculating the diameter of a set

I am trying to prove that if $X$ is complete and $ M \subset X$ has a finite epsilon net, then $M$ is relatively compact. I have the proof to hand: I don't understand how they calculated that ...
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116 views

Prove expansive function on a compact set is surjective.

Let $M$ be a compact set and $(M,d)$ be a metric space, define function $f:M\to M$ such that for all $\,p,q\in M$ $$d(f(p),f(q))\ge d(p,q)$$ Prove $f$ is surjective. I observed that compactness ...
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37 views

Given $A \subset \mathbb{R}$ is bounded, what is an open cover of $A$?

I am looking for a good example of a covering of a bounded set $A$ in $\mathbb{R}$ Currently my example is $\{I_n\} = \{(k + n - \frac{\epsilon}{2^n}, k + n - \frac{\epsilon}{2^n})\}_{n \geq 0}$, ...
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0answers
23 views

Orientation of a triangulated compact surface, using orientations of triangles

The questions I am working on asks me to :"Give the definition of an orientation of a triangulated compact surface by using orientations of triangles" I know that a surface is orientable if the ...
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2answers
39 views

Is this basic function space compact?

Let $A=L^2(X)$ be the space of square integrable functions on a compact Euclidean space $X$. If we equip $A$ with the usual 2-norm, is $A$ compact? Edit: And if we restrict AA by adding the ...
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1answer
40 views

Intersection of nested compact sets in a Hausdorff space

Suppose that a non-empty Hausdorff space $X$ is compact and there is a continuous map $f:X \to X$. Let $X_1=X$ and put $X_{n+1} = f(X_n)$ inductively for all $n \in \mathbb{N}$. Prove that $A ...
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1answer
40 views

Show that the special unitary group $SU(n)$ is a compact topological group

What I know: $SU(n)=${$A \in U(n): detA=1$} where $U(n)=${$n \times n$ matrices $A: AA^*=I=A^*A$} with elements in $\mathbb{C}$ and $A^*$ is the complex transpose of $A$ A topological group is a ...