## Proof That Open Sets Defined By Inverse Functions Constitute a Topology

Proof

Letbe a function from a setonto a topological space

The family of subsets ofdefined byis a topology on

Proof

is a topology onso

Sinceis onto,

Also

Hence

Letrepresent a family of subsets inFor each

We have

Sinceis a topology onhence

and

Sinceis a topology onand

Henceandis a topology on