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

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

Even more, from Projection from Product Topology is Continuous and Projection from Product Topology is Open $pr_i\mid_{Y_i}:Y_i\to X_i$ is an homeomorphism because it is open, continuous and bijective.