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Questions tagged [compactness]

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 $[ \sqrt 2, \sqrt 3] \cap \mathbb{Q}$ an open subset of $\mathbb{Q}$?

Consider the set of rational number $\mathbb{Q}$ as a subset of $\mathbb{R}$ with the usual metric. Let $K = [ \sqrt 2, \sqrt 3] \cap \mathbb{Q}$. I have some confusion in my mind that ...
0
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2answers
34 views

Is $\mathbb{Q} \cap (a,b)$ for irrationals $a,b$ is not compact in the $\mathbb{Q}$ with $d(x,y)=|x-y|$?

In my exam of analysis we ask to proof $\mathbb{Q} \cap (\sqrt{2},\sqrt{3})$ is not compact . i have few questions about this problem . i have a solution for the problem but beside my problem one of ...
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votes
0answers
20 views

Problem of Compactness Theorem [on hold]

There is a set of squares with partitions (walls) between some cells inside them is given. It is known that any square can be paved with these squares so that there is a path from the upper side to ...
0
votes
2answers
53 views

Proving $\mathbb{N}$ is not compact

In $\mathbb{N}$, we define the topology defined by $$T=\emptyset \cup\{\{0,1,2...,n\} :\space n\in \mathbb{N}\}\cup \mathbb{N}$$ Now I want to prove that $(\mathbb{N},T)$ is not compact. Suppose $(\...
0
votes
0answers
15 views

Let $U$ be a compact subset of $\mathbb{R}$ with the usual topology. Prove it's compact by sequences. [duplicate]

I'm stuck here. I'm trying to solve this from a "topological point of view", not as a metric space. Also I'm asking for the direction that's not solved on this question. I'll first define "compact by ...
1
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1answer
26 views

Characterize the compact subspaces in a topological space.

It’s known that every closed subset of a compact topological space is also compact. However, it’s not always true that compact subspaces are closed(by taking the cofinite topology on $\mathbb{Z}$ the ...
0
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2answers
22 views

Show that given a proper map, for each closed set F its image f(F) is closed

For my math class, I have to provide the following proof: Given two metric spaces $(X,d)$ and $(Y,\rho)$, a continuous map $f: X \rightarrow Y$ is called proper if $f^{-1}(K) $ is compact for each ...
0
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1answer
26 views

Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets

I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it. The Baire Category theorem asserts that if X is a complete metric space ...
0
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2answers
13 views

compactness and relative compactness

Suppose that $M$ is compact subset of a Banach space X. Is $M$ relatively compact too? As far as I know, there is a characterisation of compact sets via Hausdorff $\varepsilon$ - net theorem and it'...
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0answers
18 views

Space filling curves: Hilbert vs Peano

It seems that both the Peano's and Hilbert's space filling curve are same in nature. What is the basic difference between them?
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1answer
32 views

$A$ is compact and closed then prove …

I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two ...
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0answers
31 views

Some doubt regarding space filling curve.

Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square. My doubt: we need to show that ...
3
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1answer
44 views

Given an averaging operator $A: l_p \rightarrow l_p$, why $A$ is not compact

Let $A$ be an operator $A: l_p \rightarrow l_p , 1 <p<\infty$ $$A(x_1, ..., x_n, ...)=\left(x_1, \frac{x_1+x_2}{2}, ..., \frac{x_1+...+x_n}{n}, ...\right)$$ I want to show that operator $A$ is ...
-2
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1answer
31 views

continuous one-one map from [0,1] to [0,1]x[0,1] [closed]

Why a continuous one-one map from [0,1] to [0,1]x[0,1] can't have dense range? If it's range contains a ball then which arise contradiction?
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0answers
11 views

Explains to the distance of one closed and one compact set are alwasy greater than zero [duplicate]

One follow up question to this question Example to show the distance between two closed sets can be 0 even if the two sets are disjoint In this question, we get an example for A and B such that d(A,B)...
1
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1answer
33 views

A sequence of rationals converging to an irrational point (proving that $\mathbb Q$ is not locally compact)

Here is my attempt to prove that $\mathbb Q$ is not locally compact. (My questions are below the proof.) Suppose $\mathbb Q$ is locally compact. Then it is locally compact at every point. Let $x\in \...
2
votes
1answer
27 views

A cover of Locally connected space with certain compactness property

Suppose $X$ is a locally connected Hausdorff space. If $X$ is $\sigma$-compact and locally compact, is it always possible to find a countable set of precompact connected open sets $\{U_n\}$ (which ...
2
votes
2answers
115 views

A diifficulty in understanding a sentence in a paragraph in Guillemin and Pollack p.77

The paragraph is given below: But I have a difficulty in understanding the sentence starting in the forth line by "If we furthur ...." until its end, could anyone explain it for me please? thanks!
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0answers
53 views

Property of set $\left \{y \in \Bbb R : y =\lim\limits_{n \rightarrow \infty} f(x_n), \text {for some sequence }\ x_n \rightarrow +\infty \right \}$ [closed]

Let $f : \Bbb R \longrightarrow \Bbb R$ be a continuous function and $A \subseteq \Bbb R$ be defined by $$A=\left \{y \in \Bbb R : y =\lim_{n \rightarrow \infty} f(x_n), \text {for some ...
0
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1answer
12 views

Intersection of sequentially compact sets

Let A, B (non-empty) sequentially compact sets. Then the intersection of A and B is sequentially compact. One can prove this hypothesis by selecting a sequence in the intersection and observing a ...
0
votes
1answer
16 views

Would not $Z \times Y$ be limit point compact set but not compact set?

I saw that in Mukresh's book there is an example which shows that limit point compactness may not imply compactness. The example says $Z_+ \times Y$ is the space where every subset has limit point. ...
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1answer
33 views

If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.

If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed. Can anyone give me a trivial counter example?
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0answers
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Application of Ulam Lemma

I can’t understand how to derive the following conclusion from Lemma (Ulam) Let $(X, \tau)$ be a Polish space and $\mu$ a positive finite Borel measure on $X$. Then for every $\epsilon >0 $ ...
1
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1answer
26 views

Intersection of two compact subsets in a Hausdorff space

Prove that if $K$ and $L$ are compact subsets of a Hausdorff space $X$, then $K \cap L$ is a compact subset of $X$. I understand that since $K$ and $L$ are compact subsets, they each have finite ...
0
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1answer
40 views

Show (F$_{k}$) converges uniformly to some continuous function

Suppose ${0<r<1}$. For each k $\in$ $\mathbb{N}$, define F$_{k}$ $\in$ C$\bigl($[-r,r]$\bigr)$ by F$_{k}$(x) = $\sum_{n=1}^k$ x$^{n}$. Show (F$_{k}$) converges uniformly to some continuous ...
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0answers
22 views

Wallace's theorem on rectangular neighborhoods of compact rectangles

Theorem. Let $A\subset X,B\subset Y$ be compact subspaces of topological spaces $X,Y$. Let $A\times B\subset W\subset X\times Y$ with $W\subset X\times Y$ open. Then there exists opens $A\subset U\...
0
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2answers
34 views

A concern on the definition of compactness in a metric space [duplicate]

Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence. This is how I'm given the definition of a compact metric space and it confuses me. ...
2
votes
1answer
48 views

Discuss compactness of the set of $L^2$ bounded functions

Discuss weak and strong compactness of the following subsets of $L^2(0,1)$: $A=\{u\in L^2(0,1):||u||_{L^2}\le1\}.$ I know some theorems which might be helpful, but I don't know if I applied them ...
7
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2answers
161 views

A two player game on compact topological spaces

I've though of an infinite game that two players may play on a given topological space $(X,\tau)$. It goes like this. On turn $n$ Player I selects a point $x_n\in X$ and Player II selects a ...
4
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1answer
38 views

Can a noncompact metric space have a maximal metrizable Hausdorff compactification?

We know the Stone-Čech compactification $(h, \beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,\gamma X$) is any other ...
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0answers
17 views

“If and only if” condition for the relatively compactness of $S \subset c_0$

Let $S \subset c_0$. How to show that $S$ is relatively compact if and only if $S$ is bounded and $\forall \epsilon>0 \, \exists n_0 \in \mathbb N$ such that for all $n_0 \leq n$: $|x_n| \leq \...
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2answers
32 views

The closure of convex hull compact

Let $A_n=\{x_n,x_{n+1},...\}\subset E$ for each $n\in \mathbb N$, such that $E$ is a Banach space. If the closure of the convex hull of $A_n$ is compact i.e $\overline{co}(A_n)$ compact, is $\...
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1answer
33 views

Are non-orientable manifolds necessarily compact?

If not, what is an example of a non-compact, open manifold that is non-orientable? So if non-orientability $\Rightarrow$ compactness, is there a theorem and what is the proof?
2
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2answers
179 views

Is the following set a compact set?

Let $A$ be defined as $$A:=\{f\in C^1([0,1],\mathbb{R}) : \|f\|_{C^1} \leq 1\}.$$ I have shown that the set is precompact. But is $A$ a complete set? Or an other question: Is $A$ a closed set?
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1answer
10 views

Tightness of a finite measure on a complete and separable metric space

Let $(X, \rho)$ be a metric spac and let $\mu$ be a finite Borel measure on it. Assume thay $(X, \rho)$ is complete and separable. Prove that for every $\epsilon > 0$ there is a complact subset $K \...
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0answers
52 views

Cardinality of compacts with positive $(>0)$ Lebesgue measure in $\Bbb R^3$

I need to prove that it's same as $[0,1]$ (continuum). Let's say I have proved "fact" that closed balls with positive radius in $\Bbb R^3$ have same cardinality as $[0,1]$. Does it prove that ...
0
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2answers
50 views

Showing that $[0, \omega_1]$ is compact. [duplicate]

I want to show that $[0, \omega_1]$ is compact, where $\omega_1$ is the least uncountable ordinal, and I have just been introduced to the concepts of ordinals. The tips I have seen to showing this ...
2
votes
1answer
29 views

Is every continious image of a non-compact space is non-compact?

Is every continuous image of a non-compact space is non-compact? I was thinking about constant function. I think it will be false. Am I right?
0
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1answer
15 views

Intersection of a closed and a dense subset of a Compact Hausdorff space be empty?

Suppose that $X$ is a compact Hausdorff space. Let $\{X_i\}_{i \in \mathbb{N}}$ be a sequence of dense open subsets of $X$. Let $E$ be a closed subset of $X$. Can it happen that $\left(\cap_{i \in \...
0
votes
1answer
35 views

Let $A$ be the set of all rational $p$ such that $2 < p^2 < 3$.Then $A$ is choose the correct option

Let $A$ be the set of all rational $p$ such that $2 < p^2 < 3$.Then $A$ is choose the correct option $1)$ compact in $\mathbb{Q}$ $2)$closed and bounded in $\mathbb{Q}$ $3)$Not compact in $...
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1answer
41 views

Entire function bounded on a set

"Consider the set $T = \{ \alpha ∈ C|∃ a, b ∈ Z :\alpha = a + bi \}$ Let $g$ be an entire function which satisfies that $g(z + \alpha) = g(z)$ for all $z ∈ \mathbb{C}$ and all $\alpha ∈ T$. Prove ...
0
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1answer
20 views

Continuous and bounded functions and Riemann integrability

Suppose that $v=v(t,x)\in C^1([0,+\infty]\times\mathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):\mathbb{R}\to \mathbb{R}$$ is Riemann integrable over $\mathbb{R}$? Certainly $...
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0answers
21 views

invariant neighbourhood under a continuous group action

I found this statement and I am really struggling trying to come up with a proof of it. The situation is the following: Let $G$ be a compact topological group acting continuously on a compact ...
0
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1answer
17 views

Problem: Is set connected and compact

Is $ \left\{ (x,y, 1+x+y) \in \mathbb{R^3} \mid x,y \in [1,2] \cap \mathbb{I} \right\} $ connected as a subspace of $(\mathbb{I^3} , d_{2} )$? I guess it isn't connected since it is a subset of $\...
1
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1answer
78 views

Problem 2. A comprehensive course in Analysis. Barry simon. Page 239.

Definition (Baire set) Let X be a compact Hausdorff space. The Baire sets are the smallest $\sigma$-algebra containing all compacts $G_{\delta}$'s. Definition (Partition) Given an algebra , $\...
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0answers
45 views

Baire sets have arbitrarily fine refinements. Barry Simon. Problem 1 page 239.

Definition (Baire set) Let X be a compact Hausdorff space. The Baire sets are the smallest $\sigma$-algebra containing all compacts $G_{\delta}$'s. Definition (Partition) Given an algebra , $\...
3
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1answer
30 views

Applications of the lack of compactness of the closed unit ball in infinite-dimensional Banach spaces

I am writing a paper on the compactness of closed balls in Banach spaces, with particular attention paid to the following theorem Let $V$ be a Banach space over $\mathbb R$ or $\mathbb C$. The ...
1
vote
1answer
36 views

Hausdorff partial metric

Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of ...
0
votes
1answer
41 views

If $A$ and $B$ are compact subsets of $\mathbb{R}$, is then $ A × B $ compact?

If $A$ and $B$ are compact subsets of $\mathbb{R ^ n}$, is then $ A × B $ compact? I think it is because it's then closed and bounded because A and B are closed and bounded?
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votes
1answer
19 views

Proving that Sp(2N,R) is not locally compact

I'm working though Hall's Lie groups, Lie algebras, and representations and I want to show that the matrix Lie group $Sp(2N,\mathbb{R})$ is not locally compact. I've already shown that it fails to be ...