Evaluation Mapping is Injective iff Mappings Separate Points

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

Then:

$f$ is an injection if and only if $\family {f_i : X \to Y_i}_{i \mathop \in I}$ separates points

Proof

We have:

$f$ is an injection

\(\ds \iff \ \ \)

\(\ds \forall x, y \in X : x \ne y : \ \ \)

\(\ds \map f x\)

\(\ne\)

\(\ds \map f y\)

Definition of Injection

\(\ds \iff \ \ \)

\(\ds \forall x, y \in X : x \ne y : \ \ \)

\(\ds \family{ \map {f_i} x }_{i \in I}\)

\(\ne\)

\(\ds \family{ \map {f_i} y }_{i \in I}\)

Definition of Evaluation Mapping

\(\ds \iff \ \ \)

\(\ds \forall x, y \in X : x \ne y : \exists i \in I : \ \ \)

\(\ds \map {f_i} x\)

\(\ne\)

\(\ds \map {f_i} y\)

Definition of Cartesian Product of Family

\(\ds \iff \ \ \)

\(\ds \family {f_i : X \to Y_i}_{i \mathop \in I} \text{ separates points} \ \ \)

\(\ds \)

\(\)

\(\ds \)

Definition of Mappings Separating Points

$\blacksquare$