Uniform structure on subspace

Definition in terms of entourages
Suppose $$(X,\mathcal{U})$$ is a uniform space: $$X$$ is a set and $$\mathcal{U}$$ is a uniform structure on $$X$$. Suppose $$Y \subseteq X$$. The induced uniform structure on $$Y$$, denoted $$\mathcal{U}_Y$$, is defined as follows:

$$\mathcal{U}_Y = \{ U \subseteq Y \times Y \mid \exists \ V \in \mathcal{U}, \ U = V \cap (Y \times Y) \}$$.

Definition in terms of coarsest uniform structures
Suppose $$(X,\mathcal{U})$$ is a uniform space and $$Y \subseteq X$$. The induced uniform structure on $$Y$$ is the coarsest uniform structure on $$Y$$ for which the inclusion map from $$Y$$ to $$X$$ is a defining ingredient::uniformly continuous map.