Evaluation Mapping on T1 Space is Embedding if Mappings Separate Points from Closed Sets

Theorem
Let $X$ be a $T_1$ topological space.

Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.

Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.

Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.

Let $f : X \to Y$ be the evaluation mapping defined by:
 * $\forall x \in X : \map f x = \family{\map {f_i} x}_{i \mathop \in I}$

Then:
 * $f$ is an embedding


 * $\family {f_i}_{i \mathop \in I}$ separates points from closed sets