Proof of Existence of Lebesgue Number for Open Cover of a Compact Metric Space
Theorem
If
is a compact metric space and
is an open cover of
then there is
such that
for some %alpha for any
That is, a numberexists such that the
- neighbourhood
of any point
is a subset of at least one
The numberis called the Lebesgue number of the cover.
Proof
Sinceis a cover ofeachbelongs to at least one
Each
is open, hence for each
exists such that
for some
The collection of sets
is an open cover of
and sinceis compact, a finite subcover
exists.
The Lebesgue number is defined as
Let
then
Hence
for some