Factor Spaces are T5 if Product Space is T5

Theorem
Let $\mathbb S = \left\{{\left({S_\alpha, \tau_\alpha}\right)}\right\}$ be a set of topological spaces for $\alpha$ in some indexing set $I$.

Let $\displaystyle T = \left({S, \tau}\right) = \prod \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\mathbb S$.

Let $T$ be a $T_5$ space.

Then each of $\left({S_\alpha, \tau_\alpha}\right)$ is a $T_5$ space.

Proof
Let $T$ be a $T_5$ space.

Let $\left(S_\alpha, \tau_\alpha\right)$ be arbitrary.

By Subspace of Product Space Homeomorphic to Factor Space, we find that


 * $\left(S_\alpha, \tau_\alpha\right)$

is homeomorphic to a subspace of $T$.

By T5 Property is Hereditary, we then find that this subspace is also $T_5$.

Finally, by T5 Space is Preserved under Homeomorphism, we find that


 * $\left(S_\alpha, \tau_\alpha\right)$

is $T_5$.