Subspace of Product Space is Homeomorphic to Factor Space/Proof 1

Proof
Consider the restriction of the projection:
 * $\pr_i {\restriction_{Y_i} }: Y_i \to X_i$

From Projection from Product Topology is Continuous, $\pr_i {\restriction_{Y_i} }$ is continuous.

From Projection from Product Topology is Open, $\pr_i {\restriction_{Y_i} }$ is open.

$\pr_i {\restriction_{Y_i} }$ is also bijective.

.

Thus, by definition, we have that $\pr_i {\restriction_{Y_i} }: Y_i \to X_i$ is a homeomorphism.