Definition:Mappings Separating Points

Definition

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.

The family $\family {f_i}_{i \mathop \in I}$ is a family of mappings separating points if and only if:

$\forall x, y \in X : \paren{x \ne y} \implies \paren{\exists i \in I : \map {f_i} x \ne \map {f_i} y}$

In which case, the family $\family {f_i}$ is said to separate points.

Also see

• Definition:Mappings Separating Points from Closed Sets

Sources

• 1955: John L. Kelley: General Topology: Chapter $4$: Embedding and Metrization, $\S$Embedding in Cubes
• 1970: Stephen Willard: General Topology: Chapter $3$: New Space from Old: $\S8$: Product Spaces, Weak Topologies: Definition $8.13$