Definition 2

If $\mathfrak{U}_1$ and $\mathfrak{U}_2$ are two uniform structures on the same set X, $\mathfrak{U}_1$ is said to be finer than $\mathfrak{U}_2$ (and $\mathfrak{U}_2$ coarser than $\mathfrak{U}_1$) if, denoting by $X_i$ the set X with the uniform structure $\mathfrak{U}_i$ ($i = 1,2$), the identity mapping $X_1 \to X_2$ is uniformly continuous.

If $\mathfrak{U}_1$ is finer than $\mathfrak{U}_2$ and distinct from $\mathfrak{U}_2$, we say that $\mathfrak{U}_1$ is strictly finer than $\mathfrak{U}_2$ (and that $\mathfrak{U}_2$ is strictly coarser than $\mathfrak{U}_1$).

Two uniformities are said to be comparable if one is finer than the other.