Proof That Open Sets Defined By Inverse Functions Constitute a Topology

Proof That Open Sets Defined By Inverse Functions Constitute a Topology

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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