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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3
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25 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
0
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1answer
29 views

Show (0,1) is not compact [duplicate]

Let $I_n=\left(\frac{1}{n},1\right)$. Show that $(0,1)$ is not compact: show that any finite collection of $\{I_n\}$ will not cover $(0,1)$. Give me a hint.
3
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2answers
47 views

“Redundant” finite subcovering of a compact space.

Let $M$ be compact and $\mathcal{U}$ an open covering of M such that each $p \in M$ is contained in at least two members of $\mathcal{U}$. Show that $\mathcal{U}$ reduces to a finite subcovering with ...
0
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3answers
52 views

How does one show that $\{ \frac{1}{n} | n \in \mathbb{Z_{>0}}\} $is not compact in the standard topology?

How does one show that $\{ \frac{1}{n} | n \in \mathbb{Z_{>0}}\}$ is not compact in the standard topology of $\mathbb{R}$? I know this is not compact because if we take small enough intervals ...
3
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2answers
68 views

What are “finiteness” and “discreteness” when it comes to compact sets?

I recently found this answer by Qiaochu Yuan but I'm not sure what "finiteness" and "discreteness" function are in the context of compactness. I've read What does it mean when a function is finite? ...
0
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1answer
75 views

Existential Second Order Logic; Compactness and Löwenheim-Skolem

I'm looking for proofs of Löwenheim-Skolem and Compactness in existential SoL. I've spent a substantial amount of time on google, but can't seem to find anything!
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2answers
55 views

Can't figure out what's wrong with my proof

I have to decide if it possible to find a set $A\subset \mathbb{R}$ such that: $A$ is not connected nor compact but it is complete. At first, I thought it wasn't possible, and made the following ...
0
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1answer
26 views

closure of compact subspace

It is known that If $X$ is a Hausdorff space then every compact subspace of $X$ is closed. Hence closure of compact subspace of $X$ is also compact. My question: is there any a $T_1$ space $X$ such ...
0
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1answer
24 views

If $A$ is subspace of topological space $X$ is compact and closure of $A$ is not compact then $X$ is particular point topology

I am looking for a topological space $X$ which if $A\subset X$ is compact but closure of $A$ is not compact. From this Find a topological space X and a compact subset A in X such that closure of A is ...
1
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1answer
24 views

Compactness of given subsets of $\mathbb R^n$

Looking for some feedback for solutions to select exercises from a basic Analysis course. All comments welcome! Determine whether or not each subset of $\mathbf{R}^2$ is compact. Briefly justify ...
2
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1answer
37 views

Volume of a compact set, not necessarily convex

Looking through my lecture notes, I came across the notion that if a set $X\subset \mathbb{R}^n$ is compact and convex and $vol(X)=2^n$, then by choosing an $0<\epsilon <1$, then $X\subsetneq ...
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1answer
26 views

Holomorphic functions on a connected and compact domain

Consider the following theorem (see references at the end): If $X$ is a connected and compact complex manifold, then any holomorphic function $f : X \rightarrow \mathbb{C}$ is constant. What about ...
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0answers
19 views

Show that for an infinite subset M in the space s to be compact

I have to show that for an infinite subset M in the space s to be compact, it is necessary that there are numbers y1,y2,... such that for all x=(Ek(x))is an element of M, we have the absolute value of ...
0
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1answer
41 views

Using the open cover definition of compactness to show that the set of nilpotent $m \times m$ real matrices is noncompact

Is the set of nilpotent $m \times m$ real matrices compact? I found the proof of this statement, using Heine-Borel theorem on $\mathbb R^n$. Tha'ts quite good. But, is it possible to prove this ...
1
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1answer
43 views

Compactness and open sets

I have this small question, if $(E,\tau)$ is a Hausdorff space and $A,B$ two separated compact sets, how to prove the existence of two open disjoint sets $U$ and $V$ such that $B\subset V$ and ...
0
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0answers
26 views

an open subspace of locally compact is dense

Let $X$ be locally compact Hausdorff. Then a subspace $A$ of $X$ is dense and locally compact iff $A$ is open. I can prove the necessary condition. But for the sufficient condition, I can not get ...
0
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2answers
31 views

an open subspace of compact space

It is know that every compact subspace of Hausdorff space is closed and every closed set is compact. So I have a question as folows: is there any compact non-Hausdorff space $X$ such that every open ...
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0answers
20 views

Stone-Weierstrass on a sequentially compact space

I am unable to prove the Stone-Weierstrass Theorem on a compact metric space via its sequential compactness. If that were possible, one could probably prove the Theorem on a sequentially compact ...
0
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1answer
54 views

open subspace of locally compact

It is kown that A closed subspace of a locally compact space is locally compact If $X$ is locally compact Hausdorff and a dense subspace $Y\subseteq X$ is locally compact iff $Y$ is open. From the ...
0
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2answers
33 views

Compact sets and Open sets in a metric space

I have from reading up on things understood that open sets in a metric space is not compact. Though I have no clue why. I would like to know why is it they are not compact? I know that a compact set ...
2
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1answer
21 views

Annulus containing a circunference

Let $S^1=\{x\in\mathbb{R}^2\mid\lVert x\rVert=1\}$, where $\lVert x\rVert$ denotes the Euclidean norm. I am asked tho prove that if $U$ is an open set, $S^1\subset U$, then there exists ...
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2answers
43 views

For normed vectorspaces $V$, $A,B \subset V$ if $A$ is compact and $B$ is closed then $A+B$ is closed

I am looking for a 'direct' way to show the following statement: Problem: Let $V$ be a normed vectorspace, show that if $A$ is compact and $B$ is closed then $A+B:= \lbrace a+b \mid a \in A, b \in ...
0
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0answers
20 views

When is a bounded set in a metric space contained in a compact set?

If $A$ is a bounded subset of a metric space $(X,d)$ with nearest point property , then is it true that $A$ is contained in some compact set ? If $A$ is a totally bounded set of a metric space $(X,d)$ ...
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3answers
48 views

Compact normed vector space

Let $V$ be a normed vector space.If $V\neq \{0\}$ is it true that our space cannot be compact?
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3answers
39 views

Recursive use of the Axiom of Choice

In a standard proof that any sequence-compact metric space $(X,d)$ has a (finite) $\varepsilon$-net, the approach is the following: Make a sequence $(x_n)$ such that $$ x_{n+1}\notin\bigcup_{i=1}^n ...
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1answer
55 views

Theorem 4.20(c) in Baby Rudin: Is every continuous function whose domain is an unbounded subset of $\mathbb{R}$ uniformly continuous?

Here is Theorem 4.20 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Let $E$ be a non-compact set in $\mathbb{R}^1$. Then (a) there exists a continuous function on ...
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2answers
98 views

Theorem 4.20 in Baby Rudin: How is this map not uniformly continuous?

Let $E$ be a bounded, non-compact subset of $\mathbb{R}$, let $x_0$ be a limit point of $E$ such that $x_0 \not\in E$, and let $f \colon E \to \mathbb{R}$ be defined by $$f(x) \colon= \frac{1}{x-x_0} ...
7
votes
2answers
84 views

Compact set and continuous function [duplicate]

Let $(E,d), (E',d')$ be two metric space, and $f:E\rightarrow E'$ an injective function such that the image of any compact set from $E$ is compact in $E'$. How can I prove that $f$ is continuous? ...
2
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1answer
35 views

Proving a topological space is separable

I am trying to prove the following statement: Prove that if (X,d) is a compact metric space, then X must be separable. Where separable means the following: We say a topological space is separable ...
2
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0answers
48 views

Homeomorphic to $ [0,1]$?

Let $(E,d)$ be a metric space, $f~:~[0,1] \to E$ continuous such that $f$ is not constant. Is it true that $f([0,1])$ contains a subset homeomorphic to $[0,1]$?
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0answers
27 views

Arzela ascoli theorem, question?

I have a quick question, in the proof of the Arzela Ascoli theorem one uses the fact that $X$ in $C(X)$(the space of continuous function $X\rightarrow \Bbb C$) is separable. But I don't really see ...
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0answers
26 views

How to define a compactly generated space?

I engaged two definitions for a compactly generated space: http://en.wikipedia.org/wiki/Compactly_generated_space 1) In topology, a compactly generated space (or k-space) is a topological space ...
0
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3answers
36 views

Clarification of Open cover

$E$ is a compact metric space. Consider a compact set $A \subset C(E)$ where $C(E)$ denotes the set of all continuous functions on $E$. Since $A$ is compact, any open cover of $A$ has a finite ...
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0answers
40 views

If a set is closed and unbounded, is it still possible for it to be sequentially compact?

Sorry if this is a trivial question, but I couldn't find an answer for it yet. I know a set $S$ is compact iff every open cover of $S$ has a finite subcover. I also see how this is not the case for ...
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1answer
31 views

Compactness of the set of points where a continuous function achieves a local maximum

Let $(K,d)$ be a compact metric space, and $f:K\rightarrow \mathbb{R}$ be a continuous function on $K$. Define: $$M=\left \{ x\in K :\text{$f$ achieves a local maximum in $x$} \right \}$$ I need to ...
0
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1answer
25 views

Show that the compactification is the Alexandroff-compactification

In our reading we had and proved the following theorem concerning the compactification with respect to a family of bounded functions: Theorem Let $E$ be a discrete countable infinite space ...
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1answer
42 views

How do I determine if a subset of a metric space is compact?

If I have some subset of a metric space, is it always possible to determine if it is compact? If so, how? It seems to be quite easy to show something is not compact(in terms of what is required: ...
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0answers
28 views

Reference request: $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$

On a bounded domain $\Omega$, I am looking for a reference saying that $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$. I tried all the usual texts (Showalter, Evans, ...
6
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4answers
279 views

Show that there are non-well-founded models of Zermelo Fraenkel set theory

I have been working on this problem for several hours, and my understanding just isn't there. Here's what I've gathered: Using downward Lowenheim-Skolem theorem, we know that any consistent set of ...
3
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0answers
65 views

When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
0
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0answers
17 views

Integrability in closed interval on $R$ and continuity

I am studying baby version of fubini's Theorem. In the very first step I were to use uniform continuity of the function, given that the function is continuous on $R$. I think I get the uniform ...
3
votes
2answers
73 views

Are $L^\infty$ bounded functions compact in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a compact subset of $L^2$? (Compact in the topology induced by the ...
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2answers
47 views

Constract a compact set of real numbers whose limit points form a countable set. [duplicate]

This is exercise $2.13$ in Rudin. Can't we simply define such set as $[a, b]$, with all members being rational? It is bounded, and closed (proof is straightforward), and the limit points are all ...
5
votes
2answers
95 views

Interior of a compact 3-manifold

I have an orientable 3-manifold $X$, such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ How to find a compact 3-manifold $M$ such that $X= ...
6
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2answers
58 views

Metric space of infinite binary sequences

Let $\Omega = \{0,1\}^{\mathbb{N}}$ be the space of infinite binary sequences. Define a metric on $\Omega$ by setting $d(x,y) = 2^{-n(x,y)}$ where $n(x,y)$ is defined to be the maximum $n$ such that ...
0
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2answers
31 views

Proving noncompactnes by showing open cover with no finite subcover

Define: $$S = \{f \in C([0, 1],\Bbb R) : |f(x)| \le 1 \; \forall x \in [0, 1]\}$$ I have an open cover for the set $S$: $$U_{n} := \{f \in C([0, 1],\Bbb R): |f(0) − f(1/n)| < 1\}$$ for each $n ...
0
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3answers
48 views

Show that if a subset $E$ of a compact metric space $X$ is compact in $X$, then it is closed in $X$.

I am self-studying Royden's Real Analysis; Exercise 58 of Section 9.5, "Compact Metric Spaces", asks: Let $E$ be a subset of the compact metric space $X$. Show that the subspace $E$ is compact if ...
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1answer
39 views

Question about the Image of a compact transformation of a Hilbert space

$T$ is a compact operator on a Hilbert space. Show that $\operatorname{im}(T)$ does not contain a closed infinite dimensional subspace. Here is my attempt at the problem: Suppose that ...
0
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1answer
66 views

Preserved properties through continuous linear maps

I just looked at the fact (at least according to Definition 2.8.1. in Distribution Theory by Friedlander et al.) that for $K_0\subseteq{\bf R}^{n(0)}$ compact, $\Omega_1\subseteq{\bf R}^{n(1)}$ open ...
2
votes
1answer
99 views

Product of sequential sequentially compact spaces is sequential

I am trying to show that the product of two sequentially compact sequential spaces is sequential. Can someone help me? Edit: I found that there is a reference for this: Boehme T.K., Linear s-spaces, ...