Subspace topology

Definition

We define the subspace topology as follows.

Given a topological space $$(X,\mathcal{J})$$ and any $$Y\subset X$$ we can define a topology on $$Y,\ (Y,\mathcal{J}_Y)$$ where $$\mathcal{J}_Y=\{Y\cap U|U\in\mathcal{J}\}$$

We may say " $$Y$$ is a subspace of $$X$$ (or indeed $$(X,\mathcal{J})$$ " to implicitly mean this topology.