Is a projection map continuous?

Yes. A projection map is continuous.

 

Let π₁ : X × Y → X be a projection map.

Let U be open in X.

π₁⁻¹(U) = U × Y

U is open in X and Y is open in Y, so U × Y is open in X × Y.

 

Therefore, a projection map is continuous.

 

We know that a projection map is an open map.

Also, a projection map is surjective.

 

We can conclude that a projection map is a quotient map.

 

* Note that a projection map is not injective, (so the inverse doesn't exist) so it is not a homeomorphism.