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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Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
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1answer
41 views

How Can I prove the three statements are equivalent?

Let $X$ be a compact Hausdorff space and $f:X \rightarrow Y$ be a quotient map. Show that the following are equivalent: (a)$Y$ is an Hausdorff space, (b)$f$ is closed map, (c)The set ...
3
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0answers
47 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
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1answer
20 views

Distance between two compact subsets is always $\geq$ distance between two particular points of the subsets

Show that if $K_1$ and $K_2$ are compact subsets of $\mathbb R^p$, then there exist points $x_1$ in $K_1$ and $x_2$ in $K_2$ such that if $z_1$ belongs to $K_1$ and $z_2$ belongs to $K_2$, then ...
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1answer
43 views

Relative compactness of metric space

I know that in a metric space $X$ compactness, countable compactness and sequential compactness of a subspace $X'$ are equivalent using the definition of countable compactness as every infinite subset ...
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1answer
24 views

Relative compactness implies relative countable compactness?

By using the fact that compactness implies countable compactness, I think that relative compactness implies relative countable compactness in any topological space. Am I right? Thank you so much!
3
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1answer
69 views

Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
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1answer
21 views

Counterexample on weaker version of result about compact sets

The following is a very well known theorem: Let X be a metric space. $K \subset X$ is compact iff every collection $ \{ F_j \}_{j\in A}$ of closed sets with the finite intersection property in K ...
3
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1answer
29 views

“Heine–Borel” for the Sorgenfrey line [duplicate]

The Heine–Borel theorem perfectly characterizes the compact subsets of the real line $\mathbb{R}$ (with the usual metric/order topology): Heine–Borel Theorem. A subset $A \subseteq \mathbb R$ is ...
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2answers
19 views

Disjoint compact subsets of a Hausdorff space are separated by disjoint open neighborhoods

Let $X$ be a Hausdorff space and let $A,B\subseteq X$ two compact subspaces which don't intersect. Show exist $U,V\subseteq X$ open which don't intersect s.t $A\subseteq U,B\subseteq V$. I ...
2
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1answer
37 views

Proof of the Lebesgue Covering Theorem

Lebesgue Covering Theorem : Suppose $\rho =\{G_n\}$ is a covering of a compact subset $K$ of $\mathbb R^p$. There exists a positive number $\lambda$ such that if $x,y \in K$ and $|x-y| < \lambda,$ ...
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26 views

Compact/open subsets of vector space

Let $K\subset U$ be compact (resp. open) subsets of a normed vector space. Must there exist an open $V$ such that $K+V\subset U$?
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1answer
31 views

Every locally compact space is compactly generated

I am using the following definitions (from Wikipedia): A space $X$ is locally compact if every $x \in X$ has a compact neighborhood; A space $X$ is compactly generated if a subset $A \subseteq X$ is ...
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3answers
48 views

Some Questions from the proof of the result : The unit interval $\mathbb I = [0,1]$ is compact

The unit interval $\mathbb I = [0,1]$ is compact I was trying to understand the proof of the above result from my textbook which goes like as follows. However, I have a few questions in mind. Please ...
2
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2answers
113 views

Mary Ellen Rudin's proof that all metric space are paracompact

Given a metric space $(X,d)$, show that the space is paracompact. I have no idea where to begin on this, and the proofs of this I have seen have been difficult for me to understand. Can anyone offer a ...
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0answers
18 views

About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
6
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1answer
52 views

Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the Arzelà–Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
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2answers
108 views

Counter example of a locally compact topological space which is not compact

I want to show that not every locally compact topological space is compact. I have one example which I am not sure if it is a correct example which is simply the whole real line. Is it a correct ...
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2answers
46 views

Is $\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$ compact in $\Bbb R^2$?

Is this set in $\Bbb R^2$ compact: $$\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$$ I think it is compact, but the answer says not. Any help is appreciated.
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1answer
28 views

Arzela-Ascoli and compactness in $C(X), l^p, L^p$

Arzela-Ascoli and compactness in $C(X), l^p, L^p$ $C(X)$ with the uniform norm and $X$ is a compact metric space, a closed and bounded set in $C(X)$ is compact if and only if it is ...
2
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1answer
48 views

The set $\{\|f\|_\alpha \leq 1 \}$ has compact closure in $C([0,1])$

Recall the Holder norm $(0<\alpha\leq 1) $ $$\|f\|_\alpha = \max\bigg\{ |f(x)| + \frac{|f(x) - f(y)|}{|x-y|^\alpha} : x,y \in [0,1], x\neq y\bigg\}$$ I want to show that the set ...
0
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1answer
40 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
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1answer
30 views

Two definitions of compact set

I am reading parallely two books on analysis, and they have two different definitions of compact set: 1) Subset A of metric space X is called compact, if every open cover of A contains a finite ...
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3answers
38 views

Local compactness is preserved under continuous open onto mappings

If $f$ is a continuous open mapping of a locally compact space $(X,\tau)$ onto a topological space $(Y,\tau_1)$, show that $(Y,\tau_1)$ is locally compact. The definition of locally compact is ...
2
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1answer
36 views

A countable, compact KC-subspace of a hereditarily Lindelöf minimal KC-space

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
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2answers
61 views

Non-compactness of $\mathbb{R}$ with the cocountable topology

Is $(\mathbb{R},\tau_{co})$ compact where $\tau_{co}$ is the cocountable topology on $\mathbb{R}$? I have the answer of my teacher but I'd like to see another one so I can understand better how ...
3
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2answers
54 views

Infinite spaces in which all subsets are compact are not Hausdorff

Let $(X,\tau)$ be an infinite topological space with the property that every subspace is compact. Prove that $(X,\tau)$ is not a Hausdorff space. I start by supposing $X$ is Hausdorff. Then I can ...
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1answer
49 views

A continuous bijection from a Hausdorff space to a non-compact space which is not a homeomorphism

Recall the following theorem: Let $X$ be a compact space and $Y$ a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the ...
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2answers
59 views

Showing that a space is normal and not locally compact

Let $E$ be the set of all ordered pairs $(m,n)$ of non-negative integers. Topologize $E$ as follows: For a point $(m,n)\neq (0,0)$, any set containing $(m,n)$ is a neighbourhood of $(m,n)$. A set ...
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1answer
46 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
69 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 ...
2
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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 ...
3
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3answers
171 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
55 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
69 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
34 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
53 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 ...
3
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1answer
107 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: ...
4
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1answer
51 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
44 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
141 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
66 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 ...
2
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0answers
31 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
38 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; ...
2
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3answers
104 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 ...
2
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3answers
90 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 ...