0
votes
0answers
37 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
27 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
22 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
90 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
23 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
58 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
117 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
26 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 ...
10
votes
2answers
176 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
54 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
45 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
56 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
71 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
124 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
73 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
50 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
42 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
44 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
35 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
40 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
59 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
37 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
27 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
77 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
28 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 ...
3
votes
1answer
68 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
113 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
45 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 ...
4
votes
2answers
82 views

$A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but not compact?

Is it true that set $A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but is not compact. We consider space $(\mathbb{R}^2, d_C)$ where ...
1
vote
1answer
41 views

Partition and open cover in compact metric space

I'm having troubles solving this problem, any help will be appreciate :) Let $X$ be a compact metric space and for all open cover $U$ of $X$ we denote $N(U)=min\{|V| : V $subcover of $ U\}$. Let ...
2
votes
1answer
79 views

uniformly convergent sequence of functions on a compact space

There's an exercise on Kaplansky's textbook that says: Let $\{f_i\}$ and $f$ be continuous real functions on a compact metric space $M$. Prove that $f_i$ converges uniformly to $f$ if and only if the ...
-1
votes
1answer
39 views

Prove the union of compact sets is itself compact (preferably by showing for sequential compactness and then with open covers)) [closed]

Let A$_1$. . . A$_n$ be compact sets in a metric space (X,d). Prove that A$_1$ $\bigcup$ . . . $\bigcup$ A$_n$ is compact
0
votes
1answer
40 views

Compactness and Convergence of Subsequences

Let $(X,\rho)$ be a metric space. Suppose that $(x_n)_{n\in\mathbb Z_+}$ is such a sequence in $X$ that any subsequence has a further subsequence that is convergent. However, the limits of these ...
0
votes
0answers
55 views

The concept of compactness

What is the meaning of, " a metric space is compact" ? I know the formal definition but i don't really understand it's true meaning, what is the concept behind compactness ? I seems like a lot of ...
2
votes
1answer
50 views

Prove or disprove a set $F$ is closed.

This is an example in my book that talks about $F$ being precompact; Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form $$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) ...
4
votes
1answer
73 views

Regarding compactness of a space

I am trying to solve the following problem: Let $X$ be a metrizable topological space. Prove that the following statements are equivalent. (a) $X$ is compact (b) $X$ is bounded with respect to ...
2
votes
2answers
116 views

Proof that a limit point compact metric space is compact.

If $(X,d)$ is limit point compact, show it is compact. I have found multiple proofs of this that first show that limit point compact implies sequential compact, which in turn implies compactness. I ...
0
votes
1answer
127 views

If the distance between any two points is less than $1$, must $X$ be compact?

Let $X$ be a complete metric space such that the distance between any two points is less than $1$. Then is $X$ necessarily compact? Thanks in advance.
0
votes
1answer
26 views

$B(R,R)$ is not closed in the topology of compact convergence

I'm doing this exercise in Munkres book, and got no clue to solve this problem. Help someone can help me. Let $B(R,R)$ be the set of bounded functions $f: R \rightarrow R$. Prove that ...
1
vote
2answers
295 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
0
votes
2answers
42 views

Proving sequential compactness from open cover compactness.

Let $(\mathcal M,d)$ be a metric space and $A\subset\mathcal M$. The following types of compactness are equivalent: (i) Each open cover of $A$ contains a finite subcover. (ii) $A$ is sequentially ...
2
votes
2answers
34 views

Question on Compactness

Let the metric space be the real numbers with the usual distance formula. Let $E$ be an open interval from $1/8$ to $2$. Then $E$ would be compact if every open cover of $E$ has a finite cover. I know ...
2
votes
1answer
159 views

Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
11
votes
1answer
122 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
2
votes
1answer
107 views

Compactness and closed balls

Let $E$ be a compact metric space, such that $\{U_i\}_{I\in I}$ is a collection of open sets whose union is $E.$ Show that there exists $\epsilon>0$ such that any closed ball in $E$ of radius ...
2
votes
1answer
199 views

Clarification on this corollary of the Arzela-Ascoli Theorem

I am given the following corollary without proof: A family of continuous functions on a compact metric space into $\mathbb R^m$ is compact iff it is closed, equicontinuous and bounded. Does ...
4
votes
1answer
378 views

closed,bounded not compact

Hi I was asked to prove that: if $S =\{ x \in \Bbb R : d(x,0) = 1 \}$ then $S$ is a closed and bounded set. The set $S$ contains only two points: $-1,1$,(it should not be a problem to prove that is ...
2
votes
2answers
85 views

What does it mean for a set to be compact in another set?

I am given the following definition: Let $B$ be a set of continuous maps with domain a metric space $A$ and codomain a metric space $N$, and $B_x=\{f(x):f\in B\}$. $B$ is pointwise compact ...
0
votes
1answer
153 views

let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove the exist an $r>0$ such that $d(x,K)\leq r \rightarrow x \in U$

I'm stuck... I would appreciate some help let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove there exists an $r>0$ such that $d(x,K) \leq r \rightarrow x \in U$