subspace topology

Let $X$ be a topological space, and let $Y \subset X$ be a subset. The subspace topology on $Y$ is the topology whose open sets are those subsets of $Y$ which equal $U \cap Y$ for some open set $U \subset X$ .

In this context, the topological space $Y$ obtained by taking the subspace topology is called a topological subspace, or simply subspace, of $X$ .