Subspace topology

Definition

Let $(X, \tau)$ be a topological space and let $S \subseteq X$. The subspace topology on $S$ is $$\tau_S = \{ S \cap U : U \in \tau \}.$$ With this topology, $S$ is a subspace of $X$.

Remarks

The set $\tau_S$ is a topology on $S$, because intersection with $S$ commutes with unions and finite intersections. A set open in $S$ need not be open in $X$. For example, $[0, 1)$ is open in the subspace $[0, 1]$ of $\mathbb{R}$, because $[0, 1) = [0, 1] \cap (-1, 1)$. For a metric space, the subspace topology is the metric topology of the subspace metric. The subspace topology is the coarsest topology on $S$ for which the inclusion map is continuous.

Explanations

L1 human

A subset of a space inherits a topology: you cut each open set of the big space down to the subset. This is how the circle and the interval get their topologies from the plane and the line.