Proof That the Rational Numbers are Not Locally Compact With the Euclidean Topology
Theorem
Let
be the set of rational numbers in the space setr of real numbers
with the topology induced by the Euclidan metric.is not locally compact.
Proof
is obviously locally compact since for any
and any neighbourhood U of
there is a compact set
such that
Now take
and any compact set
such that
Sinceis compact, it is bounded and closed, and also contains infinitely many elements of
We can find rational numbers
and
with
Let
be an irrational number.
We can define an open cover ofas
is an open cover ofwhich has no finite subcover henceis not locally compact.