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 said to separate points from closed sets :
 * $\forall x \in X, F \subseteq X : \leftparen{x \notin F \land F}$ is closed $\rightparen{\implies \map f x \notin f \sqbrk F^-}$

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