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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0answers
37 views

Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
1
vote
1answer
26 views

Understanding this proof about the intersection of compact subsets

The following proof is theorem 2.36 from Rudin's Principles of Mathematical Analysis: Theorem: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection ...
2
votes
1answer
45 views

Understanding Rudin's proof that compact subsets of metric spaces are closed.

Rudin's Principles of Mathematical Analysis has the following definition of compact: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. ...
9
votes
7answers
495 views

What does it REALLY mean for a metric space to be compact? [duplicate]

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ...
3
votes
1answer
59 views

Compactness theorem aplication

I have one problem and I m sure that can be solved by using compactness theorem but I cant solve it. Let $T$ be an $L$-theory and $\{F_i (x) \mid i\in I\}$ family of $L$-formulas. Suppose further ...
0
votes
3answers
433 views

show that torus is compact

I am having difficuties in showing a torus is compact. Initially I wanted to use Heine-Borel theorem, but after that I realise we are not working in $\mathbb{R}^n$ space. So a simple way to show torus ...
0
votes
1answer
45 views

Give the example of compact set with infinite countable derived set [on hold]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
-1
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0answers
20 views

How to prove that $t\mapsto (\cos 2\pi t, \sin 2\pi t)$ induces the one-point compactification of $(0,1)$?

Take the unit circle $S^1 = \{(x, y) \in \mathbb{R^2}: x^2 + y^2 = 1\}$ and let $h: (0, 1) \to S^1$ be the map $h(t) = (\cos(2\pi t), \sin(2\pi t)$. The compactification induced by $h$ is the same ...
1
vote
1answer
20 views

Baby Rudin Theorem 2.41 boundedness

I'm currently reading the POMA of Rudin and I don't understand the proof of boundedness of the theorem 2.41 (p. 40) of the book. It wants to prove that if $E \subset \mathbb{R}^k$ we have that every ...
2
votes
1answer
31 views

Two disjoint compact sets in a topological group

Let $(G, \cdot )$ be a compact (Hausdorff) topological group. If $A$ and $B$ are two disjoint compact subsets of $G$, how can we show that there exists a nonempty open set $V$ such $A\cdot ...
1
vote
2answers
20 views

Cantor's Intersection Theorem

If the subsets of the compact space are already non-empty, isn't it obvious that the even the smallest subset is non-empty, and so the intersection is also non-empty because it would be the smallest ...
2
votes
1answer
42 views

Help me understand the reasoning used in the following lemma (38.1) from James Munkres' Topology.

Let $X$ be a space and $h: X \to Z$ be an embedding of $X$ in the compact Hausdorff space $Z$. There exists a corresponding compactification $Y$ of $X$ such that $H:Y \to Z$ is an embedding and equals ...
1
vote
2answers
118 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?
9
votes
9answers
402 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
3
votes
2answers
178 views

Does anyone have a proof that the intersection and union of two compact sets is compact.

I have my take on it. It is quite informal and don;t know where it would be evaluated correctly on an exam. Since the sets are compact that means for every open cover there is a finite cover. When ...
1
vote
1answer
47 views

What do you call a space whose only compact sets are finite? [duplicate]

What do you call a topological space where a subset is compact iff it's finite? Is there a technical name? For example, take the discrete topology, or the countable complement topology.
1
vote
1answer
21 views

Average integral for continuous functions with compact support

Let $f$ be a continuous function with compact support in $\mathbb{R}^n$. Show that \begin{equation} \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)\,dy = f(x), \end{equation} where $B_r(x)$ is the ...
6
votes
2answers
2k views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
1
vote
2answers
592 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
2
votes
1answer
58 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...
1
vote
0answers
251 views

Subsets of $\mathbb{R}^2$ that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
9
votes
2answers
623 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...
4
votes
2answers
136 views

Prove that closed subsets of a compact set is compact. What's wrong with this proof?

I understand other methods of achieving the result, but this was my first try. I'm not sure where my mistake is, if any. And yes, I realize that using the fact that $B$ is closed would help. For a ...
0
votes
1answer
31 views

Intersection of arbitrary union of compact subsets.

My textbooks asks to prove that arbitrary intersection of compact subsets in hausdorff space is again compact. I've kinda found the counterexample $\bigcap_{1\leq x<2} [x,3]=(2,3]$, and can't find ...
5
votes
1answer
58 views

Continuity of a “minimal distance” projection $f:(X,d) \to (K, d_{|K})$ for a compact $K \subset X$. (Hint preferred)

Let $(X,d)$ be a metric space and $K$ be a compact subset of $X$. Show that for every $x \in X$ there exists $k_x \in K$ such that $$d(x,K)=d(x,k_x)$$ Suppose that for every $x\in X$, there exists ...
2
votes
1answer
27 views

Uniformly $\beta$-continuous functions (jumps no greater than $\beta$) converge uniformly to $f$, is $f$ continuous?

Let $(X,\rho)$ be a compact metric space, we say a function on $X$ is uniformly $\beta$-continuous if, for every $\epsilon > 0$, there exists $\delta > 0$ such that if $\rho(x,y) < \delta$, ...
0
votes
2answers
52 views

Reference request for Heine-Borel theorem

I would like to know a nice reference for the Heine-Borel theorem. In a text, I have the compactness argument for the following two sets. The reference should be able to cover these two cases. ...
0
votes
3answers
94 views

Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
6
votes
3answers
186 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
1
vote
2answers
30 views

Does finite covering dimension imply local compactness?

I have a space which is not locally compact and I'm trying to see if I can say anything about the dimension of the space. I suspect that it is not finite dimensional but I have thus far been unable to ...
2
votes
2answers
42 views

Open balls with radis $>\epsilon$ in a compact metric space

In a compact metric space $(X,d)$, for a given $\epsilon>0$, if $(x_j)_{j \in J}$ is a family of points of $X$ such that the balls $B(x_j, \epsilon)$ are pairwise disjoint, does it automatically ...
1
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1answer
23 views

Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$.

Let $f$ be analytic in an open set $U \subseteq \Bbb C$ and let $K \subseteq U$ be compact. Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$ ...
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votes
1answer
33 views

it is a problem on topology [closed]

let $X$ BE LOCALLY COMPACT SECOND countable hausdorff space show that there exist a sequence {$K_n$} of compact subset such that $X=\cup K_n$ and $K_n \subset Int(K_{n+1}) $ .
-1
votes
2answers
49 views

Compactness Invariant between normed spaces

Let $X$ and $Y$ be finite dimensional normed spaces. Let $D:\X \rightarrow Y$ be an isometric isomorphism then if $X$ is compact the $Y$ is also compact. I have started by choosing a sequence in $Y$ ...
4
votes
1answer
79 views

When only eventually constant sequences are convergent?

Let $X$ be a compact Hausdorff topological space whose convergent sequences are eventually constant. Is there a description of such spaces. How ''far'' these spaces from Stonean ones?
2
votes
1answer
47 views

The assumption of nonemptiness in the theorem (3.10 from Rudin) about the intersection of nested compact sets

There is Theorem 3.10(b) in baby Rudin. If $K_n$ is a sequence of compact sets in a metric space $X$ such that $K_n\supset K_{n+1}$ and if $$\lim_{n\to \infty}\text{diam}K_n=0,$$ then ...
2
votes
1answer
55 views

Proof of the Banach–Alaoglu theorem

The Banach–Alaoglu theorem states that the closed unit ball of $B'$ (where $B'$ is the dual to a Banach space $B$ over a field) is compact in the weak* topology. I'm having trouble trying to prove the ...
4
votes
1answer
23 views

Squeezing an open set and a compact set between two sets

Let $U\subseteq \mathbb{R}^2$ be open, and $C\subset U$ be compact. Show there exists $V$ open and $D$ compact such that $C\subset V\subset D\subset U$. My attempt : For each $x\in U$ consider balls ...
0
votes
1answer
30 views

Infinite set which is Dedekind finite and Weierstrass compactness

Weierstrass compactness states that each infinite set has a limit point. Why Infinite set which is Dedekind finite with discrete metric not Weierstrass compact.
1
vote
1answer
57 views

$\beta \mathbb{R}$ is a quotient space of $ \beta \mathbb{N} $

I know that a quotient space can be thought of as being an open continuous image of a space. Therefore, it would be enough to find some map from $ \beta \mathbb{N}$ open and continuous to $ \beta ...
2
votes
1answer
43 views

Question on definition of a cover and results with compactness

My professor defines a cover of $A$ to be a collection of sets whose union is equal to $A$. I am used to this being instead a superset of $A$. Doesn't this lead to contradictions? Then, there can be ...
0
votes
1answer
24 views

Does totally bounded imply covering compact?

Let S be totally bounded. So, $\forall\epsilon>0$, $S$ can be covered by a finite number of balls of radius $\epsilon$. Now, let $\{S_n\}$ be a cover of $S$ with open sets. So, $S_n\cap S$ is also ...
1
vote
1answer
17 views

Closure of set of vectors with norm 1 in $\mathbb{C}^n$

In the proof for existence of SVD, it always says - Due to compactness, we can always find a vector $v_{1} \in \mathbb{C}^n$ such that $A\,v_{1} = \sigma_{1} \, u_{1}$. Another post explained what is ...
0
votes
2answers
43 views

Understanding the proof of “continuous image of compact set is compact”

I have a trouble understanding the proof of theorem. Theorem. If a function $f:K \rightarrow \mathbb{R}$ is continuous and $K$ is compact set, then $f(K)$ is compact too. Proof. (This is part ...
1
vote
1answer
31 views

Proving that $O(n)$ is compact

Let $O(n)$ denote the group of orthogonal matrices under multiplication. We want to show that this is set is compact. To show $O(n)$ is compact, we can use Heine-Borel and show that it is closed and ...
0
votes
1answer
29 views

Property of topologically equivalent metrics [closed]

Let $X$ be a compact topological space, and $d_1,d_2$ two metrics that induce the topology on $X$. Is it necessarily true that for every $\epsilon > 0$ there exists a $\delta > 0$ such that: ...
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2answers
65 views

Easier way to prove compactness?

Consider the topological space $(X,\tau)$ with $X=\{a,b,c\}$ and $\tau=\{\emptyset ,X,\{b\},\{a,b\},\{b,c\}\}$. This space is compact if every open cover contains a finite subcover. But there are a ...
0
votes
2answers
54 views

Weighted shift operator is Hilbert-Schmidt

If $W : \ell^2 \to \ell^2$ is the weighted shift operator defined by $$W(x_1,x_2,x_3,\ldots)=(0,x_1,\frac 12x_2,\frac 13x_3,\ldots),$$ how can I show that $W$ is Hilbert-Schmidt? If I have ...
2
votes
1answer
42 views

Is the continuous image of a compact metric space second countable?

I have been trying to determine if the continuous image of a compact metric space is second countable. I know that a compact metric space can be shown to be separable and that continuous functions ...
1
vote
1answer
43 views

How do you quickly show whether an operator is compact?

In my text on functional analysis, the author defines an operator on normed space as compact if it: continuous transforms bounded sets into relatively compact sets Okay, number 1 we can work with. ...