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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A topological space is countably compact iff every countably infinite subset has a limit point

A topological space is countably compact iff every countably infinite subset has a limit point. I'm completely stuck on this one. The book is recommending to use the fact that a space is countably ...
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All small enough subsets of a compact metric spaces are covered by one element of its open cover [duplicate]

$(X,d)$ is a compact metric space, $\{U_i | i \in I\}$ is an open cover of $X$. I need to show that there is a number $\delta > 0$, such that if $A \subset X$ and diameter of $A \le \delta$, then ...
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Stone-Cech Compactification equivalence

Let $X$ a Tychonoff space, $(e, \beta X)$ Stone-Cech Compactification and consider $(i,K)$ another compactification of $X$ that satisfies: $\forall f: X \to \mathbb R$ continuous and bounded, exists $...
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1answer
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Prove that a metrizable space is countably compact iff it is compact.

Prove that a metrizable space is countably compact iff it is compact. ($\Rightarrow$) I let $\{O_i\}$ be a countable open cover for $(X,T)$ with a finite subcover. Let $\{U_i\}$ be an uncountable ...
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1answer
27 views

If $X \cup \{\infty\}$ is compact, then $X$ is locally compact?

In a textbook I'm reading, I have a topological space $X$ (which is a subspace of $L^{\infty}(E) \setminus \{0\}$ for some topological space $E$), such that $X \cup \{0\}$ is weak*-compact (in this ...
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1answer
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Prove $\lim \sup f (x_n) = f(\lim \sup (x_n)) $ and same for $\inf$

Prove: Let $A \subset \mathbb{R}$ compact, $f: A \rightarrow \mathbb{R}$ continuous, increasing monotone and $(x_n) \subset A$. Consider. Show that $\lim \sup f (x_n) = f(\lim \sup (x_n))$ and $\lim \...
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For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by A compact Lie group $G$ is semisimple if $[\...
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23 views

C¹ function in compact and polygonal path connected implies Lipschitz

"Let $f: \Omega \rightarrow \mathbb{R}^{m}$ such that $f \in C^{1}(\Omega).$ Show that, for $K \subset \Omega$ compact and polygonal path connected, $f|_K$ is Lipschitz." I can relate the function ...
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Under what condition a normal space is compact hausdorff?

We all know that every compact Hausdorff spaces are Normal can we talk about the converse? Even normal doesn't imply compact, take $\mathbb{Q}$ with the usual topology of $\mathbb{R}$.
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Proof that $C(K)$ is a Grothendieck space for $K$ an extremely disconnected compact space.

I am looking for a proof, other than the original article by Grothendieck which is in French, that the space $C(K)$ is Grothendieck when $K$ is extremely disconnected.
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Intiution behind the theorem about locally compact Hausdorff spaces

I came across a theorem in Munkres' Topology. Theorem. Let $X$ be a Hausdorff space $X$ is locally compact iff given $x \in X$ and given a neighbourhood $U$ of $x$ there is a neighbourhood $V$ of $...
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1answer
30 views

The standard n-simplex is compact set

$ " Let \ K\subseteq \mathbb R^n \ be \ a \ compact \ set,\ then\ the\ convex \ hull \ of\ K\ is\ also\ compact \ set\ " $ In order to prove this we use that the standard n-simpex as defined ...
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1answer
32 views

is $[0,1]\backslash \mathbb{Q}$ totally bounded?

I learnt totally bounded by myself. Now, I am still trying to understand the definition and looking for counterexamples which is totally bounded but not compact. The below is some of counterexamples: ...
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1answer
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The proof of $[0,1] \cap \mathbb{Q}$ is totally bounded

I want to know whether my proof is correct. Let $\varepsilon>0$. Since $[0,1] \cap \mathbb{Q}$ is dense in $[0,1]$, then for any $x\in [0,1]$, there exists an open neighbourhood $U_x =\{y: |x-y|&...
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1answer
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Equivalence of precompactness and completely boundedness.

Definitions: The set $A\subset X$ is called completely bounded if $\forall \epsilon >0 \ \exists x_1,...,x_k \in A$ s.t. $A \subset \bigcup_{i=1}^k B(x_i,\epsilon)$. The set $A$ is called ...
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1answer
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Proof of Tychonoff's Theorem for an undergrad

In the midst of learning about compactness I come across Tychonoff's Theorem: Let $\{X_i : i \in \mathcal{A}\}$ be any collection of compact spaces. Then $\displaystyle\prod_{i \in \mathcal{A}}X_i$...
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3answers
75 views

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
28 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
71 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
72 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 C(X)=\{...
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3answers
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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 $h:Z_1\...
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1answer
21 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 $K \subset V$...
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Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [closed]

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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$K$ 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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1answer
45 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 $C([0,1])$?...
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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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1answer
38 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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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 $(f_k)_{k\in\...
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"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
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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 $\{C_{\...
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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
70 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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1answer
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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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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 $K \subset E$ exist....
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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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When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
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3answers
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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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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 R^n\...
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1answer
25 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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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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2answers
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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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$X,Y$ 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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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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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
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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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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
77 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: $T: C^{1}[0,1]\mapsto C[0,1]$ $T:...
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1answer
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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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1answer
44 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 $b\neq 0$...
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1answer
35 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 ...