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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Is Inverse of a function continuous too?

I read an example from "Principles of Mathematical Analysis" by Rudin under the section 'Continuity and Compactness'. According to the example, Let $X$ be the half-open interval $[0,2\pi)$ on the ...
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1answer
22 views

Open interval $(0,1)$ is totally bounded

Is true that an open interval $(0,1)\subseteq \mathbb{R}$ totally bounded? I think it is not true. Since there is an homeomorphism from $(0,1)$ to $\mathbb{R}$ and $\mathbb{R}$ is not totally ...
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1answer
63 views

What is a finite subcover of $[0,1]^{[0,1]}$?

According to Tychonoff's theorem, under the standard topology, $[0,1]^{[0,1]}$ is compact. However, I cannot think of a finite subcover of this space. Also, how does this reconcile with the fact that ...
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1answer
67 views

Urysohn's Lemma, Stone-Weierstrass

Let $X$ be a compact space. Show that the following statements are equivalent: a) $X$ is homeomorphic to a compact subset of $\mathbb{R}^n$ b) There are functions $f_1,\dotso, f_n\in ...
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3answers
89 views

Compactness argument in SVD existence proof

The classical proof of the existence of the SVD factorization by Trefethen and Bau reports Set $\sigma_1 = \mid\mid A \mid\mid_2$. By a compactness argument, there must be a vector $v_1 \in ...
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1answer
386 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
2
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3answers
39 views

Homeomorphic but not equivalent compactifications.

I stumbled upon the definition of equivalent compactifications which is: Two compactifications $Z_1$ and $Z_2$ of the space $X$, are said to be equivalent if there exists a homeomorphism ...
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1answer
22 views

How to prove compact set and continous? [on hold]

A f:D->C , D is open set and C is complex. If A is compact set and f is continous then f(A) is compact set. A is subset of D.
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1answer
19 views

Compact subsets of a Hausdorff space

Reviewing for qual: Let $X$ be a Hausdorff space, $K$ a nonempty compact subset of $X$, and $x \in X\backslash K$. Prove that there exist disjoint, open subsets $U$ and $V$ such that $K \subset ...
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2answers
16 views

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [on hold]

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism Requirements for a homeomorphism $f:X \rightarrow Y$: $f$ is ...
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2answers
35 views

$K$ compact metric space, is there a finite set of continuous functions that separates points in $K$?

Definition: A family of functions $\mathcal{F}$ on a set $X$ separates points in $X$ if for every distinct pair $x,y\in X$ there exists $f\in\mathcal{F}$ such that $f(x)\neq f(y)$. Let $K$ be a ...
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4answers
494 views

Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space. Can you provide a good example?
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0answers
54 views

How to prove a set is closed

Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a finite probability space equipped with a filtration, i.e an increasing sequence of $\sigma$-algebras included in $\mathcal{F}$ : $\mathcal{F}_0, ...
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1answer
38 views

Question on proof of unit ball in $C([0, 1])$ not being compact

Take the sequence $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$, but we have no subsequence of $\{f_n\}$ converging in $C([0,1])$. So the unit ball is not compact in ...
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3answers
37 views

Non-compact subsets of a metric space $(X,d)$.

I'm trying to come up with an example of a metric space $(X,d)$ such that a subset $A \subset X$ is not compact, but is closed and bounded. Essentially I want to find an example that shows that a ...
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0answers
32 views

Regularly open, co-zero sets in compact Hausdorff spaces

It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous ...
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1answer
37 views

How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
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4answers
472 views

Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
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0answers
23 views

compactness of thes sequence set

Let $S$ be a compact (in the usual topology) subset of $\mathbb R^n$, let $W = \{(q_k)_{k\in\mathbb{N}}\,\mid\, q_k\in S\}$ be the set of all the sequences taking elements in $S$, let ...
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0answers
48 views

Prove that $[0,1]$ is compact. [on hold]

Prove that $[0,1]$ is compact. Using the definition of compactness. (Not using Heine-Borel theorem)
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0answers
22 views

"Correct'' morphism extension to Nagata compactifications

Can a morphism of separated schemes of finite type over a field be extended to Nagata compactifications of the schemes preserving the closed complements? Let $\mathbf{Sch}/k$ be the category of ...
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1answer
26 views

Exercise I.7.2 in Geometry and Topology by Bredon

I'm working though the first chapter in Geometry and Topology by Glenn Bredon, and I'm stuck on Exercise I.7.2, which is related to compactness. It reads: Let $X$ be a compact space and let ...
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1answer
35 views

Hausdorff compact problem

Let $X$ a Tychonoff space and the topological immersion $e: X \to \prod_{s \in S} [0,1]$. For this other question: Show that for all compact $K$ and for all continuous function $f:X \to K$, there is ...
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1answer
68 views

Showing a metric space is not complete.

Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$. I'm trying to show that this metric space is not ...
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0answers
13 views

Characterizing functions with controlled Fourier coefficiens

It's a well known fact that an infinite dimensional Banach space $E$ is not locally compact. One may consider, at which point, is this property lost, i.e. what kind of compact sets $K \subset E$ ...
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3answers
60 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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2answers
23 views

Compactness of infinite union under these conditions

Assume I have an infinite sequence $(S_k)_{k\in\mathbb N}$ of sets $S_k\subset \mathbb R^n$, assume that all the $S_k$ are compact with respect to the topology induced by some metric $d:\mathbb ...
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1answer
24 views

Show that for all compact $K$ and for all continuous function $f:X \to K$, there is $g: \overline{e(X)} \to K$ continuous with $g \circ e = f $.

Let $X$ a Tychonoff space, $S = ${$f:X \to [0,1] : f$ continuous} and consider the topology immersion $e: X \to \prod_{s \in S} [0,1]$ where $e(x) = (f(x))_{f \in S}, \quad \forall x \in X$. Show ...
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2answers
45 views

Stone-Cech compactification of real line

I know that $[0,1]$ and a unit circle $\mathbb{S}^1$ are one-point compactifications of $\mathbb{R}$ under some suitable homeomorphism. But how does one construct the Stone-Cech compactification?
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48 views

Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets?

From this Clopen subsets of a compact metric space we know that any compact metric space $X$ contains at most countably many clopen subsets ; my question is : Does there exist a compact metric space ...
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1answer
70 views

$X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?

Let $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?
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1answer
1k views

An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.

Why is this true? I think I can find a countable union of compact sets $\cup_{k=1}^\infty X_k$ such that $\cup X_k \subseteq U$ and the lebesgue measure of $U \setminus \cup X_k$ is zero. (for any ...
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0answers
23 views

Show by using finite intersection property that( $\mathbb R$,d) is not compact.

I know that this problem is an application to the statement- ($\mathbb X$,d) is compact$\iff$Every collection of closed sets in ($\mathbb X$,d) with the finite intersection property has a non-empty ...
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0answers
28 views

compactness of a sequence space

Sorry if this question might be not well-posed, I'm very very new to topology. I have a compact set $S$ of sequences $(x_n)_{n\in\mathbb N}$ in $\mathbb R^n$ and those sequences are bounded, in the ...
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1answer
29 views

Compact subsets in Topology of pointwise convergence

First of all, I know a similar question has been asked here compactness in topology of pointwise convergence, but I am still do not know how to identify compact subsets. Given a set $X$, endowed with ...
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2answers
69 views

Why Munkres §26 Exercise 11 is nontrivial?

This is probably a silly question, but I have a trivial (most likely wrong) reading of Munkres §26 Exercise 11: Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed ...
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1answer
72 views

Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ ...
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1answer
43 views

Prove the integers in the arithmetic progression topology is not compact

I've been studying for my final exam in a general topology course, and I came upon this problem about compactness that I'm have a really tough time solving. Let $a$ and $b$ be integers, with $b\neq ...
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1answer
16 views

Issue with compactness implies boundedness proof

The proof is outlined as follows: (copied from Wikipedia but Apostol gives the same idea) If a set is compact, then it is bounded: Consider the open balls centered upon a common point, with any ...
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0answers
30 views

Composition Series Analogous to Compactness?

The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness: The Jordan–Hölder theorem also holds in the context of operator ...
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1answer
32 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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1answer
46 views

Compactification, definite function

Let $\hat{X}$ be the compactification of a Locally compact Hausdorff-space $X$. Show, that it exists an unique, continuous function $p_{\hat{X}}:\hat{X}\to X^+$, whose restriction on $X$ is the ...
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1answer
48 views

If $X$ is a non-compact metric space, can $X^n$ ever be compact?

Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$
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2answers
28 views

compactness of a set of sequences

I'm sorry if this is probably a stupid and not well-posed question but I'm really new to topology. I have two compact sets $U\subset \mathbb R ^n$ and $Y\subset \mathbb R$. Then I define $Q$ to be the ...
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3answers
66 views

Hausdorff of $X$ implies Hausdorff of $Y$ under some strange condition

Let $p:X\to Y$ be continuous surjective closed mapping s.t. $p^{-1}(y)$ is compact $\forall y\in Y$, prove that: (a) If $X$ is Hausdorff, then $Y$ is Hausdorff (b) If $Y$ is compact, then $X$ is ...
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1answer
43 views

Proof or definition of compactness in lecture notes?

I am baffled with what I am seeing. First, here's what is noted as a definition in my notes Let $X$ be a set and $A \subseteq X$. A cover of $A$ by subsets of $X$ is a family $(W_i)_{i \in I}$ of ...
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4answers
111 views

Characterization of Compact Space via Continuous Function

Let $(X,\mathfrak{T})$ be a topological space. We know that if $X$ is compact and $f:X\to \mathbb{R}$ be any continuous function then $f(X)$ is bounded since the continuous image of a compact set is ...
2
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2answers
31 views

Compactness theorem, propositional calculus

Please help me with this problem. Prove that if $\land \Phi \models \lor \Psi$ (both $\Phi$ and $\Psi$ infinite) then there exist $\phi_1,...,\phi_n$ from $\Phi$ and $\psi_1,...,\psi_m$ from $\Psi$ ...
3
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1answer
52 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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1answer
72 views

Is $\beta\omega$ hereditarily irresolvable?

$X$ is called resolvable if it can be represented as a union of two disjoint dense sets, it is irresolvable otherwise. Moreover it is hereditarily irresolvable (HI) if every subspace of X is ...