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Proof That the Plane With the Usual Topology is Second Countable

Theorem

The plane with the usual topology is second countable.

Proof

A topological spaceis second countable if a countable basis exists for the usual topology.

The topology forwith the usual metric is the set of open balls with the Euclidean metric. Takeas the set of open balls with centres whose coordinates and radii are both rational. The set rational numbers is countable, as is the cartesian products of rational numbers (the cartesian product of countable sets is countable). Henceis countable.

Hencedefined in this way is second countable.