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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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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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 ...
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1answer
44 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 ...
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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 ...
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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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36 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 ...
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19 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
46 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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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
48 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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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} ...
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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? ...
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1answer
29 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 ...
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0answers
44 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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24 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
23 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 ...
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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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37 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
25 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 ...
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1answer
20 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
39 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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26 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, ...
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4answers
264 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 ...
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60 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 ...
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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 ...
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2answers
70 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
44 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 ...
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2answers
91 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= ...
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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 ...
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28 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 ...
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35 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
38 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 ...
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1answer
64 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 ...
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1answer
91 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, ...
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Proving amalgamation property in model theory

Restate the proposition Suppose $\mathcal{M}_0$, $\mathcal{M}_1$, and $\mathcal{M}_2$ are $\mathcal{L}$-structures and $j_i ~:~ \mathcal{M}_0 \rightarrow \mathcal{M}_i, ~(i = 1,2)$ is an elementary ...
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4answers
116 views

An example of an infinite open cover of the interval (0,1) that has no finite subcover

I've been having a hard time solving this problem that I was given in class. The problem states " Give an example of an infinite open cover of the interval (0,1) that has no finite subcover." I know ...
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2answers
66 views

Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness?

The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something? ...
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1answer
60 views

Prove that the Zariski space $\text{Zar} \space (K,A)$ is compact.

I posted part of the proof from Matsumura's Commutative Ring Theory. I got stuck in the last sentence where it says "Hence the intersection of all the elements of $\mathcal{A}$ is the same thing as ...
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1answer
23 views

a problem about compactness and sequential compactness in metric space

Consider a metric space $(\Bbb N, d)$ where $d(m,n) = \frac{\vert m-n \vert} {1+\vert m-n \vert}$. Need to prove that any infinite subset $X \subset \Bbb Z$ is not compact and not sequentially ...
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33 views

How to prove that a continuous mapping from a compact, connected space..

If $ f $ is a continuous mapping from a compact, connected metric space M to the real numbers and there exists a real number s such that f(m) never equals s, then there exists a constant $ c>0 $ ...
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1answer
86 views

Is the unit ball in this sequence space compact?

I have a set $X=\{\text{complex sequences } \{x_n\}: \sup\limits_{n}\sqrt{n}\left|x_n\right|\leq 1\}$ equipped with a metric ...
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29 views

is $\delta$-compact set complete?

We define $\delta$-compact metric space as monotone union of compact sets. $M=\bigcup M_i$ ($M_i\subset M_{i+1}$), is it complete?
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1answer
38 views

Example of closed unit ball?

I am not understanding the concept of ball on a set $E$ and closed unit ball $B_1$ in $B(E)$. I need to prove or disprove by example that if the closed unit ball $B_1$ is compact or not in a metric ...
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1answer
39 views

Prove compact of a set

Could anyone help me to show that the sets $\{(x,y)|f(x,y)\le \gamma, x>0, y>0\}$ are compact for all scalars $\gamma$, for the function $f(x,y)=xy+\frac{1}{x}+\frac{1}{y}$? I think it is easy ...
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1answer
59 views

compactness or not of a Lie group

Is the Lie group generated by this Lie algebra compact or not? $$ [X_i,X_j]=0, [H_i,H_j]=f^{ijk} X_k, [X_i,H_j]=0 $$ $f^{123}>0$, and $i,j,k \in \{ 1,2,3\}$. There are 6 generators in ...
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1answer
39 views

Is this strengthening of paracompactness known?

Consider a topological space $X$. What can be said about the following property? For any open cover $\mathcal U = \{ U_i \}_{ i \in I }$ of $X$, there exists an open refinement $\mathcal V = \{ V_j ...
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1answer
41 views

True or False statements about compactness of Lie group

Several statements I like to know their True or False statements about the compactness of Lie group. Semi-simple Lie algebra: Every semi-simple Lie group generated by the semi-simple Lie algebra is ...
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82 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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1answer
97 views

In a locally compact Hausdorff space, why are open subsets locally compact?

Let $X$ be a locally compact Hausdorff space, and $A \subset X$ closed. I want to show that $X - A$ is locally compact. I have found a proof here: Open subspaces of locally compact Hausdorff spaces ...
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1answer
31 views

Non-Lipschitz homeomorphism from compact metric space to itself

Is it possible to find a compact metric space $(X,d)$ with more than one point and a homeomorphism $\varphi:(X,\tau) \to (X,\tau)$ where $\tau$ is the topology induced by $d$ such that $$(\forall N\in ...