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

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

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

From Projection from Product Topology is Open, $\operatorname{pr}_i {\restriction_{Y_i}}$ is open

$\operatorname{pr}_i {\restriction_{Y_i}}$ is also bijective.

.

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