All Questions
Tagged with metric-spaces compactness
1,185
questions
2
votes
2
answers
48
views
Is there a maximum number of disjoint balls of fixed radius I can fit into a compact metric space?
Let $(X,d)$ be a compact metric space and fix $r>0$. By sequential compactness, one may not find an infinite number of disjoint $r$-balls (sets $B_r(x):=\{y \in X: d(x,y)<r\}$) in $X$ as this ...
- 31
2
votes
1
answer
55
views
Infinite-dimensional cube $[0,1]^\mathbb N$ compact/complete under certain metrics?
Consider the infinite-dimensional cube $[0,1]^\mathbb N := \{ (x_i)_{i=1}^\infty \ | \ 0 \le x_i \le 1 \ \forall i \}.$ This space can be equipped with certain metrics. Below are two of them.
$d_\...
- 1,450
1
vote
1
answer
50
views
Two sets having strongly seperated points are strongly seperated themselves
Definition :- Two sets $A$ and $B$ are strongly seperated if there exist open neighbourhoods $U$ and $V$ such that $A \subseteq U , B \subseteq V , U\cap V=\phi$
Question :$A$ and $B$ are two compact ...
- 2,589
1
vote
1
answer
34
views
If $f:X \rightarrow Y$ is an $\epsilon$-map between compact metric spaces then there exists a $\delta > 0$ such that $diameter(f^{-1}(Z)) < \epsilon$
First a map $f: X \rightarrow Y$ is called an $\epsilon$-map if it is continuous, onto and for any $y \in Y$, $diameter(f^{-1}(y)) < \epsilon$. Now I'm trying to find a $\delta > 0$ such that if ...
- 802
3
votes
1
answer
43
views
Inverse limit of arc-like spaces is arc-like
I came across this in Nadler's book, "Introduction to continuum theory".
First, for a collection of spaces $P$, a compact metric space $X$ is said to be $P$-like if for every $\epsilon > ...
- 802
1
vote
1
answer
91
views
Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Let $(X,d)$ a metric space. Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Here's my proof so far, please check it.
Proof.
Assume that $K \subseteq ...
- 19
2
votes
2
answers
46
views
Homeomorphism from unit ball to itself preserving norm
Problem: Let $B(0,1)\subset (\mathbb R^2, \lvert \cdot\rvert_2)$ be the open unit ball and let $f\colon B(0,1)\to B(0,1)$ be a homeomorphism. Show that if $(x_n)\subset B(0,1)$ is a sequence such that ...
2
votes
1
answer
24
views
Bounding difference in coordinates of points in compact subset of $\mathbb R^2$
Problem: Let $\emptyset \neq A\subset \mathbb R^2$ be a compact subset. Show that there are points $(y_1,y_2)\in A$ and $(z_1,z_2)\in A$ such that $$y_1-y_2\leq x_1-x_2\leq z_1-z_2$$ for all $(x_1,x_2)...
2
votes
1
answer
60
views
Prove the distance between $F$, $F\subset \Omega$ , $\Omega$ is open and bounded, $F$ is closed, and $\Omega^c$ is strictly bigger than $0$
I was asked the follwing question and wondered if my attempt is correct:
Let $\Omega\subset \Bbb R^n$, be an open bounded set, let $F\subset \Omega$ be a closed and non empty subset.
Define: $d=inf_{x\...
- 131
0
votes
0
answers
73
views
Suppose that if $A$ is not compact in metric space, then $\exists \delta>0$ and closed balls $B_n$ in $A$, s.t. $d(B_n, B_m)>\delta, n\not=m$.
When I was reading a textbook about functional analysis, I notice a theorem:
In metric space, if every continuous functions on a closed set $A$ is bounded, then $A$ must be compact.
In the proof, it ...
- 141
1
vote
2
answers
39
views
Compactness of $C[J,X]$?
If $J=[0,T]\subset \mathbb{R}$, and $X$ is a Banach space? Then, is $C[J,X]$ (Banach space of all continuous functions from $J$ to $X$) a compact metric space with respect to supnorm defined as $\|x\|...
2
votes
1
answer
68
views
Characterisation of a proper map
Let $X$ and $Y$ be topological spaces. A continuous map $F:X \rightarrow Y$ is called proper if the preimage of any compact subset in $Y$ is a compact subset of $X$. I wish to understand the ...
- 219
1
vote
2
answers
64
views
A doubt on a problem involving continuous function on a compact metric space
Let $X$ be a compact metric space with metric $d$ and let $f \in C (X, X)$
and such that $d (f(a ),f(b))\ge d (a, b)$ for all $a$ and $b$ in $X.$ Show that
$d(f(a), f(b)) = d(a, b)$ for all $a$ and $...
- 779
0
votes
0
answers
44
views
Trying to understand the proof for the criterion of compactness in $l_p$ space
I have the following theorem about the criterion of compactness in $l_p$ space
For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions
are equivalent:
i) $K$-totally bounded in $(l_p,||.||...
- 249
0
votes
1
answer
44
views
Is the set compact or just relatively compact?
Verify the compactness of the following set in $C[0,1]$ with metric
sup $$F=\{x_{\alpha}\in C[0,1]: x_{\alpha}(t)=\sin\alpha t,
> \alpha\in[1,2]\}.$$
My attempt:
Consider $t_0\in [0,1]$.
$\forall ...
- 249