A Topology Formed By Continuous Functions on [0,1]
Theorem
Let
represent the set of all continuous functions on
For each
and
define
The family
forms a basis for a topology
on
Proof
Suppose
Let
and let
In fact if
then
and
and
Therefore
Theorem
Letrepresent the set of all continuous functions onFor eachand define
The familyforms a basis for a topologyon
Proof
Suppose
Letand let
In fact ifthenandand
Therefore