Proof That a Topology is the Discrete Topology if and Only if Every Point is an Open Set
Theorem
Every element of a set
is the discrete topology if and only if every point ofis an open set.
Proof
If
is the discrete topology then every element ofis an open set so that
This shows that ifis the discrete topology every point ofis an open set in
Conversely suppose that every element ofis an open setSinceis a topology it contains every union of elements of
so that if
then
Since this is true for all n and x, every collection of points inis an element ofsocontains all collections of points ofand is the discrete topology on