Subspace metric

Definition

Let $(X, d)$ be a metric space and let $A$ be a subset of $X$. The subspace metric on $A$ is the function $d_A \colon A \times A \to \mathbb{R}$ with $d_A(a, b) = d(a, b)$ for all $a, b \in A$. It is the restriction of $d$ to $A \times A$. The metric space $(A, d_A)$ is a subspace of $(X, d)$.

Remarks

The function $d_A$ is a metric, because each axiom for $d_A$ is a special case of the same axiom for $d$. For example, the interval $[0, 1]$ with $d(x, y) = |x - y|$ is a subspace of the real line. The open sets of $(A, d_A)$ are the sets $U \cap A$ with $U$ open in $X$. Thus the topology of $d_A$ is the subspace topology.

Explanations

L1 human

A subspace keeps the distances of the big space and forgets the other points. For example, the unit circle in the plane, with the Euclidean distance, is a subspace of $\mathbb{R}^2$. The distance between two points of the circle is the length of the straight chord, not of the arc.