Each basis of for a set
yields a unique topology
It is not true however, that each topology has a unique basis. Two bases
and
are said to be equivalent if both give rise to the same topology ie if
Theorem
Two bases
and
on a setare equivalent if and only if for each
and each
there is
such that
and for each
and each
there is
such that
Proof
Suppose for distinct basesandthat
then for each
and each
sinceis a basis forand
there issuch that
and 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