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 $\family {f_i}_{i \mathop \in I}$ separate points from closed sets.

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 induced by $\family{f_i}_{i \mathop \in I}$.

Then:
 * $f$ is an embedding

Proof
Let $\BB = \set{f_i^{-1} \sqbrk V : i \in I, V \text{ is open in } Y_i}$.

From Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets:
 * $\BB$ is a basis for $X$

From Analytic Basis is Analytic Sub-Basis:
 * $\BB$ is a sub-basis for $X$

By definition of a $T_1$ space:
 * all points of $X$ are closed

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

From Characterization for Topological Evaluation Mapping to be Embedding:
 * the evaluation mapping $f$ is an embedding