Equivalence of Definitions of Finer Topology

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Theorem

Let $S$ be a set.

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


The following definitions of the concept of Finer Topology are equivalent:

Definition 1

$\tau_1$ is finer than $\tau_2$ if and only if $\tau_1 \supseteq \tau_2$.


Definition 2

$\tau_1$ is finer than $\tau_2$ if and only if the identity mapping $(S, \tau_1) \to (S, \tau_2)$ is continuous.


Proof

Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping on $S$.

Then:

\(\ds \tau_1\) \(\supseteq\) \(\ds \tau_2\) Definition 1 of Finer Topology
\(\ds \leadstoandfrom \ \ \) \(\ds \forall U \subseteq S: \, \) \(\ds U \in \tau_2\) \(\implies\) \(\ds U \in \tau_1\) Definition of Superset
\(\ds \leadstoandfrom \ \ \) \(\ds \forall U \subseteq S: \, \) \(\ds U \in \tau_2\) \(\implies\) \(\ds \map {I_S^{-1} } U \in \tau_1\) as $\map {I_S^{-1} } U = U$: Definition of Identity Mapping

$\quad \; \leadstoandfrom \: \: I_S$ is continuous, by definition of a continuous mapping, which is Definition 2 of Finer Topology.

$\blacksquare$