Identity Mapping is Continuous/Metric Space
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Theorem
Let $M = \struct {A, d}$ be a metric space.
The identity mapping $I_A: A \to A$ defined as:
- $\forall x \in A: \map {I_A} x = x$
is a continuous mapping.
Proof
Let $a \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
\(\ds \map d {x, a}\) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {\map {I_A} x, \map {I_A} a}\) | \(=\) | \(\ds \map d {x, a}\) | |||||||||||
\(\ds \) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Theorem $3.4$