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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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
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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
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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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37 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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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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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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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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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
24 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
18 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
36 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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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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252 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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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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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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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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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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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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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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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
87 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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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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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
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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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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
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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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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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3answers
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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
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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
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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 ...
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Compact, sequential spaces

A compact, Hausdorff space $X$ is sequential if each for each $A\subset X$ and $x\in \overline{A}$, there exists a countable set $A_0\subset A$ such that $x\in \overline{A}_0$. I am asked to show ...
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Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
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1answer
70 views

Locally Compact Spaces: Characterizations

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
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1answer
30 views

Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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1answer
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Checking that $C_{0}(X)$ is a vector space

I am trying to prove that $C_0(X)$ is closed subspace of $C_b(X)$ (bounded continuous functions) Given, $X$ is locally compact. $C_0(X)$ is the space of all continuous functions $f:X \to F$ (field of ...
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1answer
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Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
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1answer
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What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
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Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $p^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...