Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A$ be a subset of a metric space $(X,d)$, I want to show equivalence of

(i) For every $\varepsilon > 0$ there is a finite set $E = \{ x_1, x_2, \ldots, x_n \} \subset X$ with $$ A \subset \bigcup_{i = 1}^n B(x_i, \varepsilon) $$ and

(ii) For every $\varepsilon > 0$ there is a finite set $E = \{ x_1, x_2, \ldots, x_m \} \subset A$ with $$ A \subset \bigcup_{i = 1}^m B(x_i, \varepsilon) $$

The proof of $(ii) \Rightarrow (i)$ is trivial I think, for the other direction let $\varepsilon > 0$ and $E = \{x_1, \ldots, x_n \} \subset X$ with $$ A \subset \bigcup_{i = 1}^n B(x_i, \varepsilon) $$ Without loss of generality I can assume that every $B(x_i, \varepsilon)$ contains points of $A$, so all I need to do now is select a finite collection of points $a_1,\ldots,a_m \in A$ for every $B(x_i, \varepsilon)$ such that $B(x_i, \varepsilon) \cap A \subset \bigcup_{i=1}^m B(a_i, \varepsilon)$ and then take the union for every $B(x_i, \varepsilon)$. I have an intuitive feel that this is possible and I draw some pictures which convince me that this is always possible, but i have no idea how to make this formal?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Almost solution for the implication $(i)\implies(ii)$

  1. consider $\varepsilon/2$-net $\{x'_1,\ldots,x'_m\}\subset X$.

  2. For each $x'_i$ find $a_i\in B(x'_i,\varepsilon/2)\cap A$.

  3. For each $a\in A$ we can find $k$ such that $d(a,x_k')<\varepsilon/2$, so $$ d(a,a_k)\leq d(a,x_k')+d(x_k',x_k)\leq \ldots $$

  4. The rest is clear

share|improve this answer
    
that was my idea, but it don't need to be the case that $B(x_i',\varepsilon/2)\cap A \subset B(x_i, \varepsilon)$ so i might need to select other $x_i$'s, but my problem is to show that a finite collection of points $x_i$ for every $x_i'$ is enough. –  Stefan Oct 17 '12 at 23:59
    
You do not need this inclusion. Just show that every point of $A$ situated from some $x_i$ not far than $\varepsilon$ –  Norbert Oct 18 '12 at 0:03

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.