Tagged Questions

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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1answer
10 views

Correctness of reasoning about finiteness of degree of a covering map

Let $q$ be a covering $ q \colon \mathbb{R} P^{2n} \to X$, where $X$ is path-connected. Call $V_x$ the open nbhd of $x \in X$ given by the definition of covering map. We first note that $X$ must be ...
0
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1answer
16 views

A set of differential forms, uniformly bounded with their Laplacians, is precompact in $L^2$.

Let $M$ be a compact Riemannian manifold and let $\Delta$ be a Hodge Laplacian on $k$-forms. How to show that the if the set $\{u_\alpha\} \subset C^2(M,\Lambda^k)$ of $C^2$ $k$-forms is uniformly ...
5
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3answers
29 views

Show this set is closed

As part of a proof I am writing for analysis, I need to show the following set is closed: $F_n = \{x \in \mathbb{R} \, | \,x \ge 0, ~~ 2-\frac{1}{n} \le x^2 \le 2+\frac{1}{n}\}$. My current approach ...
6
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1answer
41 views

A space which is not compact but in which every descending chain of non-empty closed sets has non-empty intersection

A topological space $X$ is compact if and only if any collection of closed sets satisfying the finite intersection property has non-empty intersection. Clearly, this implies that compact spaces $X$ ...
1
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2answers
68 views

Showing $F^{-1}(C)$ is compact when $C$ is compact.

$f : X → Y$ is a map. If f is closed, and $f^{−1}(y)$ is compact in $X$ for each $y ∈ Y$ then show that $f^{−1} (C)$ is compact in $X$ for any compact subset $C$ of $Y$ . How does the proof go ...
4
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1answer
65 views

Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$

The problem statement is: Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$ My attempt at the proof is as follows: Let $f:S^n\to\mathbb{R}^n\cup\{\infty\}$ be defined as $f(x)=h(x)$ for $x\neq p$ and ...
1
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0answers
33 views

recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). ...
2
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1answer
35 views

Proper and free action of a discrete group

In Gallot, Hulin, Lafontaine's Riemannian Geometry: Definition Let $G$ be a discrete group, acting continuously on the left on a locally compact topological space $E$. One says that $G$ acts ...
0
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1answer
62 views

A property of compact subsets of metric spaces

Let $(X,\varrho)$ be a metric space and $K\subset X$ compact. Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the ...
1
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1answer
44 views

Find a topological space X and a compact subset A in X such that closure of A is not compact.

Find a topological space X and a compact subset A in X such that closure of A is not compact. I first concluded that we must have X to be a non compact and a non Hausdorff space so that closure of A ...
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0answers
19 views

Is local compactness preserved by continuous closed onto functions? [duplicate]

I've just shown for a homework problem that if $f$ is an open continuous function from $X$ onto a $T_2$-space $Y$, and $X$ is locally compact, then $Y$ is locally compact. I wonder, does this hold for ...
3
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2answers
35 views

If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
0
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0answers
11 views

Discrete set that is compact and jordan measurable

We define a) C(0) = [0,1] b) C(n) = New set that is obtained by erasing 1/3^n section long from the middle of the remaining section in C(n-1) *If C(0) 0---------1 Then C(1) = 0---xxx---1 C(2) = ...
0
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1answer
27 views

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
23 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 ...
0
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4answers
42 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 ...
2
votes
1answer
40 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 ...
3
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2answers
43 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 ...
0
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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 ...
2
votes
5answers
52 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 ...
0
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1answer
24 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 $A\subset C^0(K)$ ...
1
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0answers
27 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 ...
1
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2answers
33 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?
2
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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 ...
1
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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 ...
2
votes
2answers
43 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
58 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 ...
0
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0answers
18 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
23 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 ...
0
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1answer
38 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 ...
0
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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 ...
2
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2answers
44 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 ...
1
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2answers
31 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).
1
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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 ...
0
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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
28 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
votes
1answer
53 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 ...
0
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2answers
37 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
vote
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?
2
votes
1answer
46 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
45 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
votes
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 $d(x)=d(x_1,x_2)=d_1(x_1)+d_2(x_2)$ ...
3
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0answers
60 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 ...
11
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3answers
230 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 ...
0
votes
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 ...
1
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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 ...