Evaluation Mapping on T1 Space is Embedding if Mappings Separate Points from Closed Sets
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Theorem
Let $X$ be a $T_1$ 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 $\family {f_i}_{i \mathop \in I}$ separate points from closed sets.
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 induced by $\family{f_i}_{i \mathop \in I}$.
Then:
- $f$ is an embedding
Proof
Let $\BB = \set{f_i^{-1} \sqbrk V : i \in I, V \text{ is open in } Y_i}$.
From Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets:
- $\BB$ is a basis for $X$
From Analytic Basis is Analytic Sub-Basis:
- $\BB$ is a sub-basis for $X$
By definition of a $T_1$ space:
- all points of $X$ are closed
Since $\family {f_i}_{i \mathop \in I}$ separate points from closed sets then:
- $\family {f_i}_{i \mathop \in I}$ separates points
From Characterization for Topological Evaluation Mapping to be Embedding:
- the evaluation mapping $f$ is an embedding
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology: Chapter $4$: Embedding and Metrization, $\S$Embedding in Cubes, $5$ Embedding Lemma
- 1970: Stephen Willard: General Topology: Chapter $3$: New Space from Old: $\S8$: Product Spaces, Weak Topologies: Theorem $8.16$