Exercise 1, § 7

Let $\mathfrak{T}_1, \mathfrak{T}_2$ be two topologies on the same set $X$. Show that $\mathfrak{T}_1$ is finer than $\mathfrak{T}_2$ if and only if every filter on $X$ which is convergent in the topology $\mathfrak{T}_1$ converges to the same point in the topology $\mathfrak{T}_2$.