Characterization for Topological Evaluation Mapping to be Embedding

Theorem
Let $X$ be a 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 induced by $\family{f_i}_{i \mathop \in I}$.

Then:
 * $f$ is an embedding


 * $(1)\quad$ the topology on $X$ is the initial topology with respect to $\family {f_i}_{i \mathop \in I}$
 * $(2)\quad$ the family $\family {f_i}$ separates points

Necessary Condition
Let $f$ be an embedding.

Sufficient Condition
Let:
 * $(1)\quad$ the topology on $X$ be the initial topology with respect to $\family {f_i}_{i \mathop \in I}$
 * $(2)\quad$ the family $\family {f_i}$ separate points

Also see

 * Characterization for Continuous Mappings Separate Points from Closed Sets


 * Evaluation Mapping on T1 Space is Embedding iff Separates Points from Closed Sets