Subspace of Metric Space is Metric Space
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$.
Let $\struct {H, d_H}$ be a metric subspace of $\struct {A, d}$.
Then $d_H$ is a metric on $H$.
Proof
By definition of restriction:
- $\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$
As $d$ is a metric, the metric space axioms are all fulfilled by all $x, y \in A$ under $d$.
As $H \subseteq A$, by definition of subset, all $x, y \in H$ are also elements of $A$.
Therefore the metric space axioms are all fulfilled by all $x, y \in H$ under $d_H$.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces