## Proof That the Set of Real Numbers With the Discrete Topology is Not Second Countable

Theorem

The set of real numbers with the discrete topologyis not second countable.

Proof

Since the the topology onis the discrete topology, ifis any real number then

setr is not countable so setr with with the discrete topology is not second countable.

Also ifis a second countable, then it is first countable. A second countable base contains a countable baseLetbe the set of elements ofcontaining some elementThenis a countable local base atandis a first countable space.