More About Functioins and Compact or Closed Sets
Many of the properties of sets are preserved by continuous functions.
Ifis uniformly continuous andis a bounded set thenis a bounded set.
Proof: ChooseThere issuch that ifthenSinceis a bounded set there aresuch that but thensinceimplies thatThus is bounded.
Letbe continuous withcompact, thenis compact.
Proof:is continuous hence uniformly continuous on a compact spaceand sinceis compact it is bounded henceis bounded. Letbe an accumulation point ofthen there is a sequenceof points insuch thatfor all n andconverges to Sincefor eachthere is a sequenceof points in E such that f(x-n) =y-n for each n. There is a subsequencethat converges to some x-0 in E . The sequence converges to f(x-0) since f is continuous at x-0 butis a subsequence ofand converges tosothereforeis closed.
This theorem has the consequence that there existsuch thatsince is a compact set and bothandbelong toso there aresuch that andso
Example:withis not compact andis not uniformly continuous. is unbounded andis not compact.
Example:withis compact andis uniformly continuous.is compact.