- #1

Fredrik

Staff Emeritus

Science Advisor

Gold Member

- 10,872

- 420

## Homework Statement

Is every subset of a totally bounded set (of a metric space) totally bounded?

## Homework Equations

F is said to be totally bounded if, for every [itex]\epsilon>0[/itex], there's a finite subset [itex]F_0\subset F[/itex] such that [tex]F\subset\bigcup_{x\in F_0}B(x,\epsilon)[/tex], where [itex]B(x,\epsilon)[/itex] is the open ball of radius [itex]\epsilon[/itex] around x.

## The Attempt at a Solution

Suppose that [itex]E\subset F[/itex], and that F is totally bounded. Let [itex]\epsilon>0[/itex] be arbitrary. We know that there exists a finite set [itex]F_0\subset F[/itex] such that [tex]E\subset F\subset\bigcup_{x\in F_0}B(x,\epsilon)[/tex], but this doesn't seem to help, since [itex]F_0[/itex] doesn't have to be a subset of E. We might even have [itex]F_0\cap E=\emptyset[/itex]. So now I'm starting to think that maybe E doesn't have to be totally bounded at all. For example, if F is some open ball in [itex]\mathbb R^2[/itex] and E is some kind of fractal or something.