Equivalence of Definitions of Finer Topology

Theorem
Let $S$ be a set.

Let $\tau_1$ and $\tau_2$ be topologies on $S$.

Proof
Let $\operatorname {id}: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping on $S$. Then:

$\: \: \: \leadstoandfrom \: \: \operatorname {id}$ is continuous, by definition of a continuous mapping, which is Definition 2.