Subspace of Product Space is Homeomorphic to Factor Space

Theorem
Let $\left \langle {\left({X_i, \vartheta_i}\right)}\right \rangle_{i \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

Consider $T = \left({X, \tau}\right) = {\displaystyle \prod_{i\in I} \left({X_i, \vartheta_i}\right)}$

Then for each $\left({X_i, \vartheta_i}\right)$ there is a subspace $Y_i\subset X$ which is homeomorphic to $X_i$.

Proof
Take any factor space $X_i$, and $z_j \in X_j$ where $j \ne i$.

Then $\displaystyle Y_i = X_i \times \prod_{i\ne j\in I} \{z_j\} \subseteq X$ is a subspace of $(X,\tau)$.

Even more, from: we have that $\operatorname{pr}_i \restriction_{Y_i}: Y_i\to X_i$ is a homeomorphism because it is open, continuous and bijective.
 * Projection from Product Topology is Continuous
 * Projection from Product Topology is Open