Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition

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Theorem

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.

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 $\family{f_i}_{i \mathop \in I}$.

Let $f$ be an embedding

Then:

$(1)\quad$ the topology on $X$ is the initial topology with respect to $\family {f_i}_{i \mathop \in I}$

$(2)\quad$ the family $\family {f_i}$ separates points

Proof

$(1)$ The Topology on $X$ is the Initial Topology

Let $f \sqbrk X$ denote the image of $f$.

Let $\tau_{f \sqbrk X}$ be the subspace topology on $f \sqbrk X$.

By definition of embedding:

$f \restriction_{X \times f \sqbrk X}$ is a homeomorphism between $X$ and $f \sqbrk X$

From Subspace of Product Space has Initial Topology with respect to Restricted Projections:

$\tau_{f \sqbrk X}$ is the initial topology on $f \sqbrk X$ with respect to the mappings $\family {\pr_i \restriction_{f \sqbrk X} : f \sqbrk X \to Y_i}_{i \mathop \in I}$

Let $\tau$ be the topology on $X$.

From Homeomorphic Topology of Initial Topology is Initial Topology:

$\tau$ is the initial topology on $X$ with respect to $\family {\pr_i \restriction_{f \sqbrk X} \circ f \restriction_{X \times f \sqbrk X} : X \to Y_i}_{i \mathop \in I}$

We have:

\(\ds \pr_i \restriction_{f \sqbrk X} \circ f \restriction_{X \times f \sqbrk X}\)

\(=\)

\(\ds \pr_i \circ f\)

Composition of Mapping with Mapping Restricted to Image

\(\ds \)

\(=\)

\(\ds f_i\)

Composite of Evaluation Mapping and Projection

Hence:

$\tau$ is the initial topology on $X$ with respect to $\family {f_i : X \to Y_i}_{i \mathop \in I}$

$\Box$

$(2)$ The Family $\family {f_i}$ Separates Points

By definition of embedding:

$f$ is a homeomorphism between $X$ and $f \sqbrk X$

By definition of homeomorphism:

$f$ is an injection

From Evaluation Mapping is Injective iff Mappings Separate Points:

the family $\family {f_i}$ separates points.

$\blacksquare$