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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4
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1answer
301 views

Is it possible to construct Hausdorff compact topology on every set?

I'd like to know if it's possible to construct Hausdorff compact topology on every set. Assume the axiom of choice if needed. Thanks for ideas.
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0answers
21 views

Sum of compact sets is compact without using continuity [duplicate]

I want to show that if $A$ and $B$ are compact sets, then $A+B$ (that is, the set $\{a+b : a \in A , b \in B\}$) is compact. I know that $A+B$ is bounded, but am having trouble showing that it is ...
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3answers
79 views

Compactness, why is $(0,1)$ not compact? I need the “thought process” [duplicate]

See, I am told that $(0,1)$ is not compact as a subspace of $\mathbb{R}$. Question is, how do I conclude that? The hint says $(\epsilon,1)_{\epsilon>0}$ does not have a finite subcover. But ...
2
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2answers
38 views

Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
1
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1answer
51 views

Property of compact metric space

Let $X$ be a compact metric space. Let $A$ be a closed subset of $X$ and let $x\in X$ be a point not in $A$. Show that there exists two disjoint open sets $U$ and $V$ such that one contains $A$ and ...
0
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1answer
25 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
0
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1answer
41 views

Continuous function on a non-compact set

I'm trying to show if $X$ is non compact ($X \subseteq \mathbb{R}$) then there is a cont function $f:X \rightarrow \mathbb{R}$ which is bounded but doesn't attain it's bounds. I'm trying it for a set ...
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4answers
89 views

Why can't a open interval in $\mathbb{R}$ be compact?

If I choose $(0,1)$ and do a covering like this: $(-1,1/2)$ and $(1/3,2)$ it's a finite covering, then $(0,1)$ is a compact set?
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2answers
75 views

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 $...
1
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1answer
50 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 ...
0
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2answers
66 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 ...
3
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1answer
61 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 ...
0
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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 $y_{n_k,m}...
0
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2answers
28 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 ...
2
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1answer
30 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 $\{ u_{...
0
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1answer
34 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$(.), ($\mu(.)$...
0
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1answer
26 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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2answers
75 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 ...
0
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1answer
29 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: ...
3
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3answers
95 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 ...
0
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1answer
40 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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2answers
37 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
35 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
66 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
39 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 ...
0
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1answer
59 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
72 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
26 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
44 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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0answers
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], \mathbb{R}^...
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1answer
23 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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2answers
18 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 ...
0
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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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2answers
47 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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0answers
68 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
42 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 $(x_n)_{n\in\mathbb{N}...
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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
19 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 \frac{1}{10}\...
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0answers
59 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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0answers
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 ...
0
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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
20 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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0answers
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...
1
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1answer
49 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 ...
1
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1answer
59 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 done ...
1
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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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2answers
54 views

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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2answers
60 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 ...