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.
$\blacksquare$