## Proof That the Restriction Function is Continuous

Letbe a continuous function.

Ifis the restriction oftothenis continuous.

Proof

For any open set

Sinceis continuous

By definition of the induced topology

Henceand sois continuous.

Letbe a continuous function.

Ifis the restriction oftothenis continuous.

Proof

For any open set

Sinceis continuous

By definition of the induced topology

Henceand sois continuous.