Proof That the Rational Numbers are Not Locally Compact With the Euclidean Topology

Theorem

Letbe the set of rational numbers in the space setr of real numberswith the topology induced by the Euclidan metric.is not locally compact.

Proof

is obviously locally compact since for anyand any neighbourhood U ofthere is a compact setsuch that

Now takeand any compact setsuch that

Sinceis compact, it is bounded and closed, and also contains infinitely many elements of

We can find rational numbersandwith

Letbe an irrational number.

We can define an open cover ofas

is an open cover ofwhich has no finite subcover henceis not locally compact.