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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The family of open intervals that do not contain $0$

Let $T$ be the collection of all open sets in $\mathbb{R}$ not containing $0$ union $\mathbb{R}$ i.e $$T=\{(a,b)\subset\mathbb{\bar R}:0\notin(a,b)\}\cup\{\mathbb{R}\}$$ Then what is true about $T$? ...
2
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1answer
18 views

Proving the pre-image of $[0,1]$ is sequentially compact under this continuous function?

I'm given that to prove that $f:\mathbb R^n\rightarrow \mathbb R$ is continuous and that $\forall u\in \mathbb R^n,$ $f(u)\geq \|u\|.$ I'm then supposed to show that $f^{-1}([0,1])$ is sequentially ...
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4answers
40 views

In proving “if a set is compact, then it must be closed”, why does the finite subcover behave differently than the infinite open cover?

Proof from http://en.wikipedia.org/wiki/Heine-Borel_theorem: '''If a set is compact, then it must be closed.''' Let $S$ be a subset of $\mathbb{R}^n$. Observe first the following: if $a$ is ...
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1answer
34 views

Construct a Compact set of Real numbers whose limit points form a countable set

What do you think about this set? K = 0 $ \cup$ {1/n : n $\epsilon$ Natural numbers} It has one limit point which is zero, so it is countable and it is compact. Is this correct? Also, do you ...
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2answers
38 views

Proof verification of compactness

Let $K$ be the set $\{0\} \cup \{1/n : n \text{ is an element of the positive integers}\} $ Prove that $K$ is compact. In my head, it seems that what they are asking in this question to prove is ...
3
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2answers
90 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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0answers
13 views

Prove the uniform convergence of series in a compact space

I have to prove that if $\sum\limits_{n=1}^\infty \frac{1}{|a_n|}$ converges, then $\sum\limits_{n=1}^\infty \frac{1}{x-a_n}$ converges absolutely and uniformly in any compact space that $\forall n ...
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5answers
41 views

Well-ordered set with greatest element is compact

Let $X$ be a well-ordered set with a greatest element $\alpha$. We consider all sets of the form $]x,y]$ where $y \in X$ and $x$ is either another element of $X$ or the symbol $\leftarrow$ (the ...
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1answer
21 views

Necessity in Arzela-Ascoli theorem

I am trying to prove necessity of boundedness and equicontinuity in Arzela-Ascoli and I don't know how to go about it. More precisely,I have: Let $K$ be a compact metric space, and $A\subset C^0(K)$ ...
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0answers
22 views

On Compact and Measurable Sets with Positive Length

Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure ...
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2answers
32 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
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3answers
34 views

The nonemptiness of the intersection of compact sets such that all finite intersections are nonempty

From Rudin's Principles of Mathematical Analysis: Theorem 2.36: If {$K_\alpha$} is a collection of compact sets of a metric space X such that the intersection of every finite subcollection of ...
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0answers
10 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
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2answers
40 views

Show that the set is compact using the definition

The set in question is $\{0\}\cup \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ (for $n\in\mathbb N$). Okay, so for a set to be compact, every open cover of it must be able to be broken down into a ...
2
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1answer
17 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
3
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3answers
56 views

Proving that $ω_1$ is locally compact

I'm trying to show that $ω_1$ is locally compact, but when doing so, I need to show something else, which got me a bit stuck on. I'm taking a $\alpha\in ω_1$, so $\{\alpha\}$ is an open set. Since ...
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0answers
17 views

Question about subcovers

I was just wondering if I have an open cover consisting of open sets $U_1$ and $U_2$, is $\{U_1, U_2\}$ a finite subcover? That is, does the subcover have to be proper (cannot contain all of the ...
2
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1answer
21 views

$A \subset M$ is sequentially compact $\iff$ every infinite subset of A has an accumulation point in $A$

Another analysis homework problem here: Show that $A \subset M$ is sequentially compact $\iff$ every infinite subset of A has an accumulation point in $A$. I have two questions: 1) For the ...
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1answer
34 views

Prove that the set of accumulation points of a bounded subset of Rn is compact.

We are allowed use of the Heine-Borel theorem that states that a set is compact in $\mathbb{R}^n$ iff the set is closed and bounded. I know that the set of accumulation points is closed, but I am ...
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2answers
20 views

Compactness of subset of $\mathbb{R}$

The following is a problem from my analysis homework: Let $A$ be an infinite set in $\mathbb{R}$ with a single accumulation point in $A$. Must $A$ be compact? What I'm having trouble understanding ...
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2answers
42 views

topological properties of a given set

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have ...
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2answers
30 views

Rudin - Exercise 12, Cap. 2 Principles of Mathematical Analysis

Let $\Bbb{K}\subset\Bbb{R}$ consist of $0$ and the numbers $\frac{1}{n}$, for $n=1,2,3,\dots$. Prove that $\Bbb{K}$ is compact directly from the definition (without using the Heine-Borel theorem).
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1answer
31 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
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1answer
30 views

Small question about the existence of an homeomorphism

I need to prove this small fact to understand the proof of a theorem: Suppose $X, Y$ are countable discrete subespaces of $\beta \omega$, $Y \subset X$, and let $p \in \beta \omega \setminus \omega$ ...
2
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1answer
27 views

Continuous function on $\mathbb{R}^{n}$ preserving compactness - some clarification

My professor went over a proof of the following in class: Suppose $A \in \mathbb{R}^{n}$ is compact and $f:A \rightarrow \mathbb{R}^{n}$ is continuous. Then $f(A)$ is compact. The proof ...
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3answers
27 views

Open Cover of Compact Set Minus a Point on the Boundary

I am having a hard time thinking of an infinite (uncountable or not) open cover of a compact set missing a point on its boundary in $\Bbb R^2$, so that the open cover has no finite subcover. I know ...
4
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1answer
55 views

Is the following set compact

Let $S$ be the set $S = \{(e^{-x}\cos (x), e^{-x}\sin(x)) : x\geq 0\} \cup \{(x,0):0\leq x \leq 1\}$. Is $S$ compact? I know I have to show that if $S$ is closed and bounded, then it is compact. ...
0
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1answer
52 views

compactness of $L^2$ normed space

I have no idea how, where to start. I mean that we can show the compactness of the set via existence of convergent subsequence. But how can I take it? Please give a clue. This is my problem Show ...
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2answers
35 views

Proving that a compact space is regular

I am studying a book and I am stagnating on a what should be a straightfoward proof: Show that if $X$ is compact, $V\subset X$ is open and $x\in V$, then there exists an open set $U$ in $X$ with ...
1
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1answer
29 views

Is the embedding $L^2(0,T;H^1) \subset L^2(0,T;L^2)$ compact?

Is the embedding $L^2(0,T;H^1(\Omega)) \subset L^2(0,T;L^2(\Omega))$ compact?
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1answer
43 views

Compact set and its extreme points

I am reading Chapter 3: Convexity of Rudin's "Functional Analysis". Here is the problem I'm having trouble solving (number 18): Let $K$ be the smallest convex set in $\mathbb{ R}^3$ that contains ...
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3answers
39 views

R with usual topology and Z with discrete topology are homeomorphic?

I can't decide if the reals with the usual topology is homeomorphic to Z with the discrete topology. I know know that there must be uncountable sets in both topologies because the power set of z has ...
2
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2answers
47 views

$K_i$ is compact for $i=1,2$ implies $K=K_1 \times K_2$ is compact

I have to prove the following: We have two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ and their product-space $X=X_1 \times X_2$ with metric $d=d_1 \times d_2$ (so $d(x)=d(x_1,x_2)=d_1(x_1)+d_2(x_2)$ ...
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0answers
59 views

Compact subsets of $L^\infty$

The Riesz Frechet Kolmogorov theorem gives a necessary and sufficient condition for a subset of $L^p(\Omega)$ spaces for $1\leq p<\infty$ and equipped with Lebesgue measure to be relatively compact ...
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3answers
228 views

Spaces where all compact subsets are closed

All compact subsets of a Hausdorff space are closed and there are T$_1$ spaces (also T$_1$ sober spaces) with non-closed compact subspaces. So I looking for something in between. Is there a ...
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1answer
24 views

Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. [duplicate]

Define $A(K) = \{f : K \rightarrow \mathbb{R}$ $| f$ is continuous$\}$. $K$ is compact. Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. Since a Banach space is complete then every Cauchy ...
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1answer
24 views

multiplication of compact sets

There are $ A $ and $ B $ subsets of $ \mathbb{R} $, defined $ AB = \{ ab: a \in A, b \in B\} $. Now suppose that $ A $ and $ B $ are compact sets, then prove that $ AB $ is a compact set. I took a ...
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0answers
44 views

Question on topology and Zorn's lemma

I am having trouble showing a paracompact cover has a local refinement (that part is by definition) which admits another cover indexed by the same set such that each open set in the new set has ...
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1answer
28 views

Compactness of the convergent to zero sequences

I've gotta prove that $$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$ is ...
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0answers
25 views

How to prove that a function is compact (closed and bounded)?

The specific function I am looking at is $f(x_1,x_2) = x_1x_2 + \frac 1{x_1} + \frac 1{x_2}$, where for a fixed $a > 0, f(x) \le a$ and $(x1,x2) > 0 $ I'm really just looking for where to ...
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0answers
14 views

Examples of monotone mappings?

I am looking for some interesting (non-trivial) examples of functions between normal spaces which are perfect and monotone, i.e., functions which are surjective and closed preimages of singletons ...
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0answers
27 views

Help with continuum theory

A continuum is a compact connected Hausdorff space (sometimes metric is included in the definition). I have yet to find any references that help me understand composants and components of a ...
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1answer
28 views

Compactness, Convexity, Convex Hull of Sets including sequences

Is the following set compact, is it convex and what is the convex hull? $V = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n :\frac{1}{1 + i} \leq x_i \leq \frac{1}{i}, i=1,2,...,n\}$ My thoughts: I was ...
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2answers
116 views

Is compactness a generalization of completeness

Is the concept of compact spaces a generalization of completeness to non-metric topological spaces?
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2answers
293 views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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1answer
55 views

Proving budget constraint is compact.

Given the prices $p \in \mathbb{R}_{+}^{k}$ and income $y \geq 0$, define the consumer's budget set as the set of feasible consumption bundles: $\beta(p,y) = \{x \in \mathbb{R}_{+}^{k}: ...
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0answers
35 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
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0answers
41 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
3
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0answers
73 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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0answers
47 views

Proof of Propositional Compactness Theorem

I am going through the proof for the following form of compactness theorem. Statement: If Φ is an unsatisfiable set of propositional formulas, then some finite subset of Φ is unsatisfiable -- ...