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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*Continuous real function on a compact set is bounded and achieves greater value and less in that set.* [on hold]

Continuous real function on a compact set is bounded and achieves greater value and less in that set. I know how prove is done and how boundeness achieve greater value, please help prove someone as ...
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1answer
45 views

Boundedness of continuous functions on compact sets

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum. What happens if we replace $K$ by a closed ...
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2answers
64 views

If a property holds for arbitrary compact set in a metric space, does it also holds for the metric space?

Suppose a metric space $(X, d).$ Further suppose that a property $A$ holds for arbitrary compact subset of $X.$ Does the property $A$ also hold for $X$? Context I hoped for some general theorems of ...
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1answer
30 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 ...
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3answers
168 views

In a non-Hausdorff space, can a compact subset fail to be closed?

In a Hausdorff space $X$, every compact subset $Y$ is closed. So if I relax the condition on $X$ being Hausdorff, is it possible compact subset $Y$ of $X$ not being closed?
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1answer
49 views

Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...
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0answers
19 views

In a normal family, for every $\epsilon>0$ there are finitely many functions $f_{j}$ such that $\min_j\sup|f-f_j|<\epsilon$ for every $f$

Let $f_{n}$ be a normal family. Why does there exist finite many indices $f_{n_{1}}, \ldots, f_{n_{k}}$ such that $\{f_{n}: n = 1, 2, \ldots\} \subset \bigcup_{j = 1}^{k}\{f: |f(z) - f_{n_{j}}(z)| ...
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2answers
63 views

Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
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1answer
32 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
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2answers
52 views

A question on the purpose of the condition on hausdorff to prove homeomorphism

This is a theorem proved in Munkres. Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism. I knew Y being hausdorff which will be good to ...
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1answer
101 views

Can a function have a strict local extremum at each point?

A problem given in Spivak's Calculus text is to show that a function $f:[a,b]\to \mathbb{R}$ cannot have a strict local maximum at each point. I will sketch the proof below the fold. My question is: ...
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1answer
48 views

Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
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1answer
41 views

Proving that a quotient space is compact but not Hausdorff

Let ∼ be the equivalence relation on $\mathbb{R^2}$ defined by $(x, y) ∼ (x_0 , y_0 )$ if and only if there is a nonzero $t$ with $(x, y) = (tx_0 , ty_0$ ). Prove that the quotient space ...
2
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4answers
135 views

Fixed point for a continuous function on a compact set?

If $f:X \rightarrow X$ is continuous and X is compact, will $f$ have a fixed point? We know that a contraction will have a fixed point but I have not come across an example of a continuous function ...
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2answers
64 views

What are some compact (Hausdorff) groups?

I just realized today that I don't know any compact groups that aren't profinite groups or Lie groups. Generalizing from these, a product of compact groups is again a compact group, a closed ...
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3answers
58 views

Sequential compactness in $\mathbb{R}$

Well known result: Suppose $f:\mathbb{R}\to \mathbb{R}$ is continuous and let $K$ be a compact set. Then, $f(K)$ is compact. I can prove this using the definition of compactness (finding a ...
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0answers
17 views

Prove that a continuous one-to-one function from a compact space onto a hausdorff space is a homeomorphism [duplicate]

Some lecture notes I'm reading use the following lemme: let $ f : X \to Y$ be a continuous one-to-one function from a compact topological space $X$ onto a hausdorff space $Y$. Then $f$ is a ...
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2answers
34 views

Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
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0answers
30 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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1answer
34 views

Alexandroff one-point compactification = Freudenthal end point compactification / set of ends

By spaces I mean locally compact, $\sigma$-compact, connected, locally connected, Hausdorff topological spaces. I need a reference (not a proof, I already have one -- or at least I think so; ...
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3answers
101 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
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3answers
86 views

Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis

I am not able to fill the gap in proof of following theorem which is stated as... Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an ...
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1answer
34 views

Compactness and Arithmetic Confusion

Let $T$ be some theory capable of arithmetic and construct a provability predicate (which we will call $Prb_T$). Let $\mathbb{N} \models T$. Expand our language to include a new constant symbol $c$. ...
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1answer
82 views

Suppose $K$ is compact. What other types of coverings must have finite subcovers?

Let $X$ be a topological space. Call $S\subseteq X$ an $\mathcal{O}$-set if there exists an open set $O$ such that $O\subseteq S \subseteq \overline{O}$. Suppose $X$ is compact. Is it true that any ...
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1answer
195 views

Stone Cech compactification homeomorphism implies realcompactification homeomorphism

I was wondering: If $\beta X$ is homeomorphic to $\beta Y$, is it true that $\nu X$ is homeomorphic to $\nu Y$? Notation: If $f: X\rightarrow \mathbb R$, we denote it's extension by $f^\alpha: \beta ...
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2answers
87 views

Continuous function from R to a compact set

I know that a continuous function maps compact sets into compact sets. My question now is, are there continuous functions $f:{\mathbb R}\rightarrow I$, with $I=[a,b]$ ($a\neq b$)?
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1answer
32 views

zero-sets of $\beta X$

I'm trying to understand the following proof from Walker: Proposition. The zero-sets of $\beta X$ are countable intersections of closures in $\beta X$ of zero-sets of $X$. Proof. If $Z$ is a zero ...
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16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
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1answer
43 views

Prove that a set is compact

Let $X$ be a compact space, let $U$ be an open set in $X$, Let $f:U\to [0,1]$ be a continuous map. Prove that the set $$K=\{(x,t): x \in U , 0 \leq t \leq f(x) \} \subset X \times [0,1]$$ is compact. ...
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2answers
27 views

Uniform convergence on compact sets allows switching the limit and the integral.

Why does uniform convergence on compact sets allows switching the limit and the integral?
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5answers
79 views

If $A$ and $B$ are compact subset of $\mathbb R$ , then so is $A+B$.

Prove the following: If $A$ and $B$ are compact subset on $\mathbb R$ , then so is $A+B:= \{a+b\mid a\in A ,b\in B\}$. I was actually thinking about first proving that if $A\subseteq \mathbb R$ is ...
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66 views

Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$. I already appreciate your hints/answers. Thanks
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0answers
18 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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2answers
211 views

How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
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A question on $\sigma-$compact spaces

Let $A$ be a closed, $\sigma-$compact subspace of $X$ such that the quotient space $X/A$ is $\sigma-$compact. Can we deduce that $X$ is $\sigma-$compact?
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1answer
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Closed kernel in a compact group is open

The way I think it should work is that $${\rm ker} = \bigcap_{g \notin {\rm ker}} (G - g\,{\rm ker}),$$ with each $G - g\,{\rm ker}$ open. Since $G$ is compact, there should, in fact, only be ...
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22 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
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1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
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1answer
49 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
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89 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
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120 views

A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
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1answer
34 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
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1answer
47 views

Are the family of functions $C^0(I,[0,1])$ equicontinuous?

I searched but couldn't find. Are the family of continuous functions $C^0(I,[0,1])$ equicontinuous for the finite interval $I\subset\mathbb{R}$? To claim this, I guess for every $\epsilon>0$ ...
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3answers
60 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
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1answer
46 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
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1answer
57 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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1answer
39 views

Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...
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2answers
54 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
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54 views

Sequence of compact sets

Let $(X,d)$ be a metric space and consider an increasing sequence $A_n$ of its subsets such that $A = \bigcup_n A_n$ is compact. Can it happen that $A\setminus A_n$ is compact for all finite $n$?
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3answers
78 views

Show that a map with some properties is closed

Let X be a topological space and Y hausdorff and local compact. Let $f:X \rightarrow Y$ be a continuous map such that $f^{-1}(K)$ is compact for all compact sets $K$. Show that $f$ is a closed map. ...