Definition:Mappings Separating Points from Closed Sets

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 from closed sets :
 * $\forall x \in X, F \subseteq X : \leftparen{x \notin F \land F}$ is closed $\rightparen{\implies \exists i \in I : \map {f_i} x \notin f_i \sqbrk F^-}$

where $f_i \sqbrk F^-$ denotes the closure of $f_i \sqbrk F$.

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

Also see

 * Definition:Mappings Separating Points