Proof That if a Countable Local Base Exists at a Point of a Topological Set Then a Nested Local Base Exists at the Point

Theorem

If a countable local baseexists at a pointof a topological set then a nested local base exists at the point.

Proof

Let

Each setis the intersection of open sets containingso is open and contains

Also B_1 supset B_2 supset B_3 ,,,

Let U be an open set containing x then k in setn exists such that B_k subset A_k subset U.

Henceis a nested local base at