0
votes
2answers
65 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 ...
1
vote
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
votes
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 $X_2$ ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 $r(x)$ ...
8
votes
2answers
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 ...
0
votes
2answers
58 views

In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
1
vote
2answers
27 views

Proving compactness of the extended complex plane

Prove that $(\overline{\mathbb C}, \overline{d})$ with $\overline{d}(z,z')=d(\phi(z),\phi(z'))$, where $d$ denotes the euclidean distance in $\mathbb R^3$ and $\phi$ is the inverse of the ...
0
votes
0answers
39 views

How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
0
votes
3answers
57 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
4
votes
1answer
44 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
1
vote
1answer
49 views

Cardinality of all compact metric spaces

I`m looking for cardinal number of all compact metric spaces. I know that: Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set) Compact metric space is ...
0
votes
1answer
48 views

Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
1
vote
1answer
49 views

Distance between any two points in a compact metric space

I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ...
0
votes
0answers
35 views

How to prove that a sub-space of the functions $f: X \to Y$ is equicontinuous?

Let $X$ and $Y$ be two metric and compact spaces, and $C(X,Y)$ - the metric space of the continuous functions $f:X\rightarrow Y$. Denote by $Y^X$ the space of all functions (not just continuous) ...
0
votes
0answers
30 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
1
vote
0answers
31 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
1
vote
1answer
17 views

Continous function on compact interval - bounded

Let $K$ be a compact interval in $\mathbb{R}$. Then every continous function $\phi :K\rightarrow \mathbb{R}^d$ is automatically bounded. Is this a consequence of; the image of a compact is compact ? ...
1
vote
1answer
16 views

Help undestanding compactness with convergent subsequences

One way to define compactness in metric spaces is to note that in compact metric space each sequence has a convergent subsequence. Understanding compactness is difficult for me from this ...
0
votes
1answer
51 views

How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an ...
0
votes
0answers
42 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
1
vote
0answers
30 views

Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
0
votes
1answer
26 views

Metric space and compactness

Prove that if in a metric space all closed balls are compact, a subset is compact if and only if it is closed and bounded. Attempt: If all closed balls are compact, then there is a converging ...
0
votes
2answers
123 views

theorem compactness and Hausdorff

I have this theorem "$X$ is compact $\leftrightarrow\exp X$ is compact", but i can not find source of it. It concerns Hausdorff metric.
1
vote
2answers
35 views

Space of probability measures total bounded?

I want to consider a space of probability measures on some set $\Omega$. It's complete (am I right?). But I don't know whether it's total bounded. Actually, I want to prove that the space of ...
0
votes
1answer
81 views

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
0
votes
2answers
231 views

Difference between closed, bounded and compact sets. [closed]

Can somebody explain the difference between compact, bounded and closed sets with examples?
2
votes
0answers
34 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
13
votes
2answers
417 views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
0
votes
1answer
61 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
3
votes
1answer
46 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
0
votes
1answer
61 views

Compact features

Consider this problem: Let $X$ be a metric space, $U$ be open, $K$ compact and $K\subset U$, show that there exists a $r>0$ such that $B(k,r)\subset U$ $\forall k\in K$ Here $B(k,r)=\{x\in X ...
0
votes
0answers
119 views

Metric-space, counterexample in Arzela-Ascoli Theorem

My book has very few examples, so I would like an example covering this. The theorem is stated as follows. "Let $(X,d_{X})$ be a compact metric space. A subset K of $C(X,\Re^{m})$ is compact if and ...
3
votes
2answers
143 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
-4
votes
1answer
76 views

Show that the set $K=[0,1]$ is compact in $\Bbb{R}$

Please help me for the follow example. Please. The example is: Show that the set $K=[0,1]$ is compact in $\Bbb{R}$. Thanks for your solve. Thanky very much once again for your answers and your ...
-1
votes
1answer
53 views

**Are there infinite metric spaces which have no infinite compact subsets**

Please if anyone can help me in solving this example: Are there infinite metric spaces which have no infinite compact subsets If there are, please, that they look like, please help me, thank you, ...
0
votes
0answers
51 views

Is an epsilon-net dense in its totally bounded set?

By definition, a totally bounded set A possesses an epsilon-net for every epsilon greater than 0. Does this mean that every point of A is either a limit point of the epsilon-net or a point in the net? ...
1
vote
1answer
55 views

Show that these subsets of R are sequentially compact

I have to show that a) $[2, 2\frac{1}{2}] \cup [3, 3\frac{1}{3}] \cup [4, 4\frac{1}{4}] \cup ...$ b){1, 2, 3, ..., $N$} for some $ N \in \mathbb{N} $ are sequentially compact. I know that in a ...
1
vote
2answers
42 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
3
votes
1answer
52 views

Continuous function and nested compact spaces

Let $X,Y$ be metric spaces and $f:X \to Y$ be a continuous function. Let $K_n \subset X$ be a compact subspace of $X$ for $n \in \mathbb N$ such that $K_{n+1} \subset K_n$. Prove that $f(\bigcap_{n ...
2
votes
1answer
67 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
2
votes
1answer
42 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
0
votes
1answer
30 views

Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
4
votes
2answers
84 views

Show that A=$\{(x_1,…x_n) \in \Bbb R | -1\le x_1\le x_2\le …x_n\le 1\} \subset \Bbb R^n $ is closed.

The full question was: Show that A=$\{(x_1,...x_n) \in \Bbb R | -1\le x_1\le x_2\le ...x_n\le 1\} \subset \Bbb R^n $ is compact, but I was able to show correctly that it is bounded. However my ...
1
vote
3answers
133 views

How to show the intersection of two compact subsets is compact

Let (X,d) be a metric space and A,B $\subset$ X be two compact subsets. Show $A\cap B$ is also compact. I attempted this question by showing the intersection is bounded and closed. But I stated ...
1
vote
1answer
32 views

Demonstrate that the following metric space is not compact

Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact. I know that sequentially compact and ...
4
votes
1answer
77 views

Metrizable topological space $X$ with every admissible metric complete then $X$ is compact

How to prove: If $X$ is a metrizable topological space and every admissible metric on $X$ is complete then $X$ is compact. I was trying with an idea of contradiction and thereby to construct ...
6
votes
1answer
124 views

Maurice Frechet's 1904 Definitions of Compactness

I'm writing a small paper on the history of compactness. Frechet wrote in French, and I don't speak French, so I've been consulting this paper: Taylor, A.E. On page 244, I read that Frechet proved ...
0
votes
1answer
57 views

Each open cover of a sequentially compact metric space has Lebesgue number

I want to query, whether I'm right. (I'm sorry if don't use the correct words in my translation, please feel free to correct, and give me hints.) I have a metric space $(X,d)$ which is sequentially ...