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 defined by:
 * $\forall x \in X : \map f x = \family{\map {f_i} x}_{i \mathop \in I}$

Then:
 * $f$ is an embedding


 * the topology on $X$ is the initial topology with respect to $\family {f_i}_{i \mathop \in I}$

Necessary Condition
Let $f$ be an embedding.

Let $f \sqbrk X$ denote the image of $f$.

Let $f \sqbrk X$ be given the subspace topology.

By definition of embedding:
 * $f$ is a homeomorphism between $X$ and $f \sqbrk X$

Also see

 * User:Leigh.Samphier/Topology/Characterization for Continuous Mappings Separate Points from Closed Sets


 * User:Leigh.Samphier/Topology/Evaluation Mapping on T1 Space is Embedding iff Separates Points from Closed Sets