Definition 3

Given two topologies $\mathfrak{T}_1, \mathfrak{T}_2$ on the same set $X$, we say that $\mathfrak{T}_1$ is finer than $\mathfrak{T}_2$ (and that $\mathfrak{T}_2$ is coarser than $\mathfrak{T}_1$) if, denoting by $X_i$ the set $X$ with the topology $\mathfrak{T}_i$ ($i = 1, 2$), the identity mapping $X_1 \to X_2$ is continuous. If in addition $\mathfrak{T}_1 \neq \mathfrak{T}_2$, we say that $\mathfrak{T}_1$ is strictly finer than $\mathfrak{T}_2$ (and that $\mathfrak{T}_2$ is strictly coarser than $\mathfrak{T}_1$).

Two topologies, one of which is finer than the other, are said to be comparable.

The criteria for a mapping to be continuous (no. 1, Definition 1 and Theorem 1) give the following proposition: