Evaluation Mapping is Injective iff Mappings Separate Points
Jump to navigation
Jump to search
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
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 |
$\blacksquare$