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
Sinceis a bounded set there are
such that
but then
since
implies that
Thus is bounded.
Theorem
Letbe continuous withcompact, thenis compact.
Proof:
is continuous hence uniformly continuous on a compact spaceand sinceis compact it is bounded henceis bounded. Let
be an accumulation point of
then there is a sequence
of points insuch 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 toso
thereforeis closed.
This theorem has the consequence that there exist
such that
since is a compact set and both
and
belong toso there are
such that
and
so
Example:
with
is not compact andis not uniformly continuous. is unbounded and
is not compact.
Example:
with
is compact and
is uniformly continuous.
is compact.