Margin between a compact subspace and an including open subspace
In Chapter 31 (“Compact-Open Topology”), the fact that, on the set of the continuous mappings the second topology (the uniform convergence topology) is finer than the first (the compact-open topology) is said to be “clear” but actually needs some non-trivial proof. Part of this is the lemma that I state and prove below.
For any and real
, let
be the open ball centered on
and with radius
. We know that an open ball is an open subset.
The lemma says:
Let
be a metric space,
an open subspace of
and
a compact subspace of
. Then there exists
such that for all
in
,
.
Proof:
For each , since
, there exists an
such that
.
The collection of all with
is an open cover of
. Hence there exists a finite
such that
covers
.
Since is finite, there exists a real
such that for all
,
.
Let be an eventual point of
.
Since covers
, there exists
such that
, that is,
.
Let be an eventual point of
.
We have and
, which leads to
, that is,
. But
is in
and
was taken such that
. Hence
.
Hence .
We thus have such that
.