Let be a metric space. Let . Let

by .

Then is continous.

Proof.

Let . Let be a basic neighborhood of in

. Then , for some .

Let , . Then .

Also, , so

.

Therefore, is continuous.

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Let be a metric space. Let . Let

by .

Then is continous.

Proof.

Let . Let be a basic neighborhood of in

. Then , for some .

Let , . Then .

Also, , so

.

Therefore, is continuous.

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