Tagged Questions

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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34 views

A set A is compact, is its boundary compact?

I am trying to understand the concept of a boundary, and I have seen it defined $Bd(A) = \overline{A} \cap \overline{A^{\complement}}$. I was wondering three things, First how can I show that the ...
2
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2answers
60 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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0answers
15 views

Compactification of topological spaces

Is there anybody here who can gives me links of pages about compactification topological spaces ? It would be better if the pages are pdf
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2answers
30 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
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1answer
40 views

if $X$ has a finite number of isolated points, is $X$ compact?

If every real valued continuous function on $X$ is uniformly continuous is $X$ is compact? Moreover if $X$ has a finite number of isolated points, is $X$ compact now? I think that the answer to the ...
2
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1answer
37 views

$X $ is complete if every real valued continuous function on $X $ is uniformly continuous

If every real valued continuous function on $X $ is uniformly continuous,then is $X$ complete? My attempt:let $x_n$ be a Cauchy Sequence in $X$. Let $f$ be a real valued continuous function. To show ...
0
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1answer
37 views

show a subset of $\mathbb R^n$ is compact if it is closed and bounded

Use the two lemmas to prove that a subset of $\mathbb R^n$ is compact if it is closed and bounded. Lemma 1: A closed subset of a compact space is compact Lemma 2: If $X$ and $Y$ are compact then ...
2
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2answers
19 views

Compactness of two equivalent metric spaces

Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$). The following are ...
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1answer
65 views

What does it mean for a set to be compact? (intuitively)

I'm having trouble intuitively understanding what it means for a set to be compact. I know that by definition a set is compact if for every open cover of the set there exists a finite subcover. But I ...
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1answer
87 views

How can I prove that something is an open cover?

I want to prove that the union of some intervals forms an open cover for some segment. Any ideas on how to do this?
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2answers
82 views

Would a connected space contain a compact subspace

I am trying to prove that in a connected space - $X$ , for every two elements of $X$ - say $a,b$ I can find a subspace of $X$ ( say $X'$ ) , such that$ X'$ contains a,b and is also connected, and ...
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1answer
37 views

$C_b(X)$ is non-separable for $X$ non-compact

If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$. I was trying like this, but got stuck at middle: Take a ...
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1answer
30 views

Examples of non-compact connected spaces with the property…

I am looking for a non-compact connected space $X$ such that for any two disjoint closed $A,B\subseteq X$ there exists a proper closed connected $C\subseteq X$ such that $A\cup B\subseteq C$. I ...
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2answers
35 views

Proof with compact sets in Hausdorff space

Prove that every compact set in Hausdorff space is closed. Let $(X,\tau)$ be Hausdorff space and $A,B$ compact, disjont subsets of $(X,\tau)$. Prove that exist two disjoint sets $V,W$ open in ...
1
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1answer
40 views

Differentiable Manifold minus point not compact

Suppose $X$ is an $n$-dimensional for differentiable manifold for $n \geq 1$: in our definition this is a second countable Hausdorff space with a maximal differentiable atlas. If $p \in X$ is a ...
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0answers
37 views

Does there exist a non-compact connected space with this property?

I am looking for a non-compact connected space $X$ with the following property: Any two disjoint closed subsets of $X$ are contained is disjoint complements of closed connected subsets of $X$. To be ...
0
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1answer
30 views

Showing compactness of complete metric space

I need to show that for $K>0$, $$X=\{f:[0,1]\rightarrow [0,1]\mid |f(x)-f(y)|\leq K|x-y|\ \forall x,y \in [0,1]\}$$ with the metric $d(f,g)=\max|f(x)-g(x)|$ , (supremum metric), is a compact ...
1
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2answers
55 views

Give an example of a set that is closed but not compact nor bounded. Prove your answer.

Let $X = (0,\infty)$ with the usual topology in $\mathbb{R}$ and the the usual metric. Consider $A \subset X$ where $A = [1, \infty)$. Then $A$ is closed as $A' = (0,1) \subset X$. My attempt is as ...
2
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1answer
28 views

Closedness of Continuous Mappings from Compact Metric Space to Compact Metric Space

Let $(X, \rho_{X})$ and $(Y, \rho_{Y})$ be two compact metric spaces. Consider the metric space $(M_{XY}, \rho)$, where $M_{XY}$ is the set of any mappings from X to Y and $\rho(f,g) := \sup_{x \in ...
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2answers
38 views

prove: a complete metric space $X$ is compact if and only if …

Let $X$ be a complete metric space. Suppose that for any infinite subset $A$ of $X$ and for any $\epsilon>0$ there are $x_1,x_2 \in A$ such that $d(x_1,x_2)< \epsilon$. Show that $X$ is ...
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2answers
38 views

continuity of a map on a $T_2$ space

Let $X$ be a $T_2$ space .Let $f:X\rightarrow \mathbb R$ be such that $\{(x,f(x):x\in X\}$ is compact.Show that $f$ is continuous My attempt: Let $x_n$ be a sequence in $X$ converging to $x$.To ...
0
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1answer
27 views

Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.

Any compact subset of $\mathbb{R}_{l} $ must be a countable set. Consider the open cover {[n,n+1): n is an integer} of R which has no subcover. So R is not compact with respect to lower limit (or ...
1
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3answers
60 views

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact.

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact. Disproof by counterxample? Not true. Let $X = \mathbb{R}$ with the usual topology and $A = (-\infty,0)$. ...
0
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1answer
39 views

Problem of a compact space

Let $X$ be a compact $T_2$ space.Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional.Show that $X$ is finite. Spent nearly 3 hours on this problem.Cant ...
7
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2answers
39 views

Give an example of an infinite compact set $A$ such that its supremum is not a limit point

I got this one on a quiz the other day (We're only working in the reals). My solution was $$A=[0,1]\cup\{3\}$$ The closed interval has the infinite points, and $\sup A=3$ is not a limit-point since ...
1
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1answer
45 views

Proof with set compactness with river metric

We have got $d_r$ metric $$d_r(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\ |x_2| + |y_2| + |x_1-y_1|, & \text{if $x_1 \neq y_1 $} \end{cases}$$ Prove that inside ...
0
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2answers
60 views

$A \subset \mathbb{R}^n$. If every continuous function $f: A \rightarrow \mathbb{R}$ is is bounded and attains its bounds then A is compact.

I'm doing a metric spaces course and got stuck on proposition. I have a feeling that I want to show that $A$ is bounded and closed then use Heine-Borel theorem. The proposition states that $f$ is ...
1
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1answer
47 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
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0answers
20 views

Principal ideal rings are not FO axiomatizable

A ring $R$ is a principal ideal ring if it is a ring and a model of $\forall I[I \text{ is an ideal} \to \exists x \forall y(y \in I \leftrightarrow \exists z (y = z*x))]$. How can one prove that this ...
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2answers
34 views

Let A and B be disjoint closed subsets of Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}. Show that if A={a} is a singleton, then d(A,B)>0.

Let A and B be disjoint closed subsets of Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}.r I have to show that if A={a} is a singleton, then d(A,B)>0 and I have no idea how to do this.
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1answer
39 views

Show that the set of uniformly Lipschitz functions vanishing at $0$ is compact in $C[0,1]$

The question is: For $K$ and $\alpha$ fixed, show that $\{f\in \operatorname{Lip}_k \alpha : f(0) = 0\}$ is a compact subset of $C[0,1]$. I was going to attempt this by using by Arzela-Ascoli theorem ...
0
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4answers
39 views

Show $K$ is Compact.

I'm new to Real Analysis and didn't know how to start this question: Let $x_n \rightarrow x$ in $(M, d)$. Let $K = \{x\} \cup \{x_n : n\in\mathbb{N}\}$. Show that $K$ is compact. I wanted to try ...
5
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3answers
55 views

Motivation for the Definition of Compact Sets

I'm currently taking my first course in real analysis, and was recently introduced to the following definition of compact sets: A set $S \subseteq \mathbb{R}$ is compact if and only if every open ...
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1answer
22 views

compactness criterion for random variables in L2

Suppose $X_n$ is a sequence of random variables such that their second moments are uniformly bounded. I would like to know a compactness criterion for this case. In analysis, if $K$ is a bounded ...
1
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1answer
16 views

Compactness of the sum of line segments

Let $A\subset(0,\infty)$. Now $X(A)\subset\mathbb{R}^2$ will be a sum of closed intervals connecting points $(0,-1)$ and $(a,\frac{1}{a})$, $a\in A$. I am asked to prove $X(A)$ is compact $\iff A$ is ...
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3answers
53 views

Compact set in R that is not convex?

Just need an example. For example, the I know the set [0,1] is compact because it is obviously closed and bounded. But I have no idea how to test for convexity
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2answers
23 views

Would the following proof be wrong? (About the intersections of compact subsets)

Let $X$ be a topological space, and let $\{K_\alpha\}_{\alpha\in A}$ be a family of closed compact subsets of $X$. Show that $\bigcap_{\alpha\in A} K_\alpha$ is compact. Proof: Let $\mathcal{T}$ ...
1
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1answer
34 views

Kelley's topology : A topological space X is compact iff each nest of closed non-void sets has a non-void intersection.

Recall that a nest is a family of sets which is linearly ordered by inclusion. This problem is from kelley's "general topology" problem 5.H. the necessity follows from the finite intersection ...
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1answer
37 views

Show that the unit sphere with centre $0$ in $\mathbb{R}^d$ is compact.

Namely, the sphere is $\{x\in\mathbb{R}^d: \| x\|_2=1\}$. I am going about this by proving that the sphere is bounded and closed. I have proved that it is bounded and I can see that it must be closed ...
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1answer
13 views

Correctness of reasoning about finiteness of degree of a covering map

Let $q$ be a covering $ q \colon \mathbb{R} P^{2n} \to X$, where $X$ is path-connected. Call $V_x$ the open nbhd of $x \in X$ given by the definition of covering map. We first note that $X$ must be ...
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1answer
17 views

A set of differential forms, uniformly bounded with their Laplacians, is precompact in $L^2$.

Let $M$ be a compact Riemannian manifold and let $\Delta$ be a Hodge Laplacian on $k$-forms. How to show that the if the set $\{u_\alpha\} \subset C^2(M,\Lambda^k)$ of $C^2$ $k$-forms is uniformly ...
5
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3answers
29 views

Show this set is closed

As part of a proof I am writing for analysis, I need to show the following set is closed: $F_n = \{x \in \mathbb{R} \, | \,x \ge 0, ~~ 2-\frac{1}{n} \le x^2 \le 2+\frac{1}{n}\}$. My current approach ...
6
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1answer
48 views

A space which is not compact but in which every descending chain of non-empty closed sets has non-empty intersection

A topological space $X$ is compact if and only if any collection of closed sets satisfying the finite intersection property has non-empty intersection. Clearly, this implies that compact spaces $X$ ...
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2answers
70 views

Showing $F^{-1}(C)$ is compact when $C$ is compact.

$f : X → Y$ is a map. If f is closed, and $f^{−1}(y)$ is compact in $X$ for each $y ∈ Y$ then show that $f^{−1} (C)$ is compact in $X$ for any compact subset $C$ of $Y$ . How does the proof go ...
4
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1answer
67 views

Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$

The problem statement is: Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$ My attempt at the proof is as follows: Let $f:S^n\to\mathbb{R}^n\cup\{\infty\}$ be defined as $f(x)=h(x)$ for $x\neq p$ and ...
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0answers
45 views

recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). ...
2
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1answer
38 views

Proper and free action of a discrete group

In Gallot, Hulin, Lafontaine's Riemannian Geometry: Definition Let $G$ be a discrete group, acting continuously on the left on a locally compact topological space $E$. One says that $G$ acts ...
0
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1answer
65 views

A property of compact subsets of metric spaces

Let $(X,\varrho)$ be a metric space and $K\subset X$ compact. Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the ...
1
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1answer
46 views

Find a topological space X and a compact subset A in X such that closure of A is not compact.

Find a topological space X and a compact subset A in X such that closure of A is not compact. I first concluded that we must have X to be a non compact and a non Hausdorff space so that closure of A ...
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0answers
20 views

Is local compactness preserved by continuous closed onto functions? [duplicate]

I've just shown for a homework problem that if $f$ is an open continuous function from $X$ onto a $T_2$-space $Y$, and $X$ is locally compact, then $Y$ is locally compact. I wonder, does this hold for ...