Definition:Metric Subspace

Definition

Let $\struct {A, d}$ 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 $H$.

That is, let $\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$.

Then $d_H$ is the metric induced on $H$ by $d$ or the subspace metric of $d$ (with respect to $H$).

The metric space $\struct {H, d_H}$ is called a metric subspace of $\struct {A, d}$.

Also see

• Subspace of Metric Space is Metric Space

• Results about metric subspaces can be found here.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Subspaces
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Definition $7.1$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.5$
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): One: Metric Spaces: $1$: Open and Closed Sets