## Proof That a Cartesian Product of Bases is a Basis For The Product

Theorem

Ifis a basis for a a topological spaceandis a basis for a a topological spacethen is a basis for a the topological space

Proof

Supposeis open. Thenis the union of cartesian productswhereandare open inrespectively hence

is a basis ofandis a basis forso for eachandwhere

Hence

The setis the required basis for