Subspace of Product Space is Homeomorphic to Factor Space/Proof 2

Proof
For each $i \in I$, let $p_i = \operatorname{pr}_i {\restriction_{Y_i}}$.

By Projection from Product Topology is Continuous and Restriction of Continuous Mapping is Continuous: $p_i$ is continuous.

Let $U \in \upsilon$.

Then by the definition of the subspace topology:
 * $\exists U' \in \tau: U = U' \cap Y_i$

Thus:
 * for each $y \in Y_i$, there exists a finite subset $I_y$ of $I$

and:
 * for each $k \in I_y$, there exists a $V_k \in \tau_k$

such that:
 * $\displaystyle y \in \bigcap \operatorname{pr}_i^{-1} \left({V_k}\right) \subseteq U'$