Definition:Mappings Separating Points from Closed Sets
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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 if and only if:
- $\forall x \in X, F \subseteq X : \leftparen {x \notin F \land F}$ 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
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 $8.13$