Proof That a Topology is the Discrete Topology if and Only if Every Point is an Open Set

Theorem

Every element of a setis the discrete topology if and only if every point ofis an open set.

Proof

Ifis 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 ofso that ifthenSince 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