Proof That the Rational Numbers are Not Locally Compact With the Euclidean Topology
Letbe the set of rational numbers in the space setr of real numberswith the topology induced by the Euclidan metric.is not locally compact.
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.