## More About Functioins and Compact or Closed Sets

Many of the properties of sets are preserved by continuous functions.

Theorem

If is uniformly continuous and is a bounded set then is a bounded set.

Proof: Choose There is such that if then Since is a bounded set there are such that but then since implies that Thus is bounded.

Theorem

Let be continuous with compact, then is compact.

Proof: is continuous hence uniformly continuous on a compact space and since is compact it is bounded hence is bounded. Let be an accumulation point of then there is a sequence of points in such that for all n and converges to Since for each there is a sequence of points in E such that f(x-n) =y-n for each n. There is a subsequence that converges to some x-0 in E . The sequence converges to f(x-0) since f is continuous at x-0 but is a subsequence of and converges to so therefore is closed.

This theorem has the consequence that there exist such that since is a compact set and both and belong to so there are such that and so Example: with is not compact and is not uniformly continuous. is unbounded and is not compact.

Example: with is compact and is uniformly continuous. is compact. 