## Equivalent Bases

Each basis of for a setyields a unique topologyIt is not true however, that each topology has a unique basis. Two basesandare said to be equivalent if both give rise to the same topology ie if

Theorem

Two basesandon a setare equivalent if and only if for eachand eachthere issuch thatand for eachand eachthere issuch that

Proof

Suppose for distinct basesandthatthen for eachand eachsinceis a basis forandthere issuch thatand vice versa.

A simple example of an bases are the bases

1. the set of all open discs produced by the metric

2. the set of all open ovals produced by the metric