1. So I worked on this problem, only to find that the solutions claim a fairly simple answer. The question is as follows(adsbygoogle = window.adsbygoogle || []).push({});

Let X be a metric space with metric d; let A [itex]\subset[/itex] X be nonempty. Show that if A is compact, d(x,A)=d(x,a) for some a[itex]\in[/itex]A.

2. Relevant equations

d(x,A)=inf{d(x,a)|a[itex]\in[/itex]A}

3. The attempt at a solution

So the solutions claimed an easy fix. The function is continuous in both variables so a continuous image of a compact set is compact and so on the set {x} × A, it reaches a minimum.

My solution was more convoluted, because I did not immediately see that the distance function was continuous in the second variable. This is how I did it.

Pf: Consider the collection, ℂ, of all sets ℂ_{ε}, ε>d(x,A) , such that a[itex]\in[/itex]ℂ_{ε}if and only if d(x,a)<ε.

Then, for any arbitrary finite collection {C_{εi}}, we can order them by set inclusion and we see that their intersection is nonempty, for if it were empty, then this would imply that d(x,A)≠inf{d(x,a)|a[itex]\in[/itex]A}.

Then, by the finite intersection property of a compact set, the intersection of all of these sets must be nonempty, so let p be that element in the intersection. Then I claim that d(x,p)≤d(x,a) for all a[itex]\in[/itex]A, for if not then d(x,p)>d(x,a) for some a, so then we can fit an ε between d(x,p) and d(x,a) (by the order properties of ℝ). Then, the infinite intersection will not contain p, a contradiction.

So, then d(x,p)≤d(x,a) for all a[itex]\in[/itex]A. So, by the definition, d(x,A)≤d(x,p) but similarly, d(x,p)≤d(x,A) since it is a lower bound of the set. Therefor d(x,A)=d(x,p).

There are a few places where I was lax on the rigor (like claiming my formation of the collection is actually ok), but let me know if it looks right.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Question about compactness

**Physics Forums | Science Articles, Homework Help, Discussion**