Topological Product with Singleton

Theorem
Let $T_1$ and $T_2$ be topological spaces.

Let $a \in T_1, b \in T_2$.

Let $T_1 \times T_2$ be the topological product of $T_1$ and $T_2$.

Then:
 * $T_1$ is homeomorphic to the subspace $T_1 \times \left\{{b}\right\}$ of $T_1 \times T_2$
 * $T_2$ is homeomorphic to the subspace $\left\{{a}\right\} \times T_2$ of $T_1 \times T_2$

A Generalization
Let $X_a$ be a topological space for each $a \in I$ and let $\displaystyle X = \prod_{a \mathop\in I} X_a$. Let $p_a$ be the $a$th projection for each $a \in I$. Suppose that $x_0 \in X$ and $k \in I$.

Let $S = \left\{{x \in X: x_a = x_{0a} \text{ when $a≠k$}}\right\}$

Then $S$ is homeomorphic to $X_k$