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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Proving budget constraint is compact.

Given the prices $p \in \mathbb{R}_{+}^{k}$ and income $y \geq 0$, define the consumer's budget set as the set of feasible consumption bundles: $\beta(p,y) = \{x \in \mathbb{R}_{+}^{k}: ...
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30 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
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26 views

weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then T is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B \in ...
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0answers
58 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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34 views

Proof of Propositional Compactness Theorem

I am going through the proof for the following form of compactness theorem. Statement: If Φ is an unsatisfiable set of propositional formulas, then some finite subset of Φ is unsatisfiable -- ...
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44 views

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
42 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 ...
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52 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
23 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
45 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
26 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!
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1answer
72 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 ...
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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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0answers
27 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 ...
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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 $x ∈ K$ ...
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1answer
53 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
109 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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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
29 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
50 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 ...
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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
31 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
39 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 ...
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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
62 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 ...
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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 ...
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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
173 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)| ...
3
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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
45 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
146 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
67 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 ...