Be $\tau_{1}$ and $\tau_{2}$ topologies over $\mathbb{N}$ Be $\tau_{1}$ and $\tau_{2}$ topologies over $\mathbb{N}$ defined by:
$\tau_{1}=\{\{m\in \mathbb{N}:m<n\}:n\in \mathbb{N}\}\cup \{\mathbb{N}\}\}$ and $\tau_{2}=\{A\subseteq \mathbb{N}: 0\in A\}\cup \{\emptyset\}$.
Check who are these topologies and prove which is the finest.

If $U\in\tau_1$, then since $0\in U$, or $U=\emptyset$,  $U\in\tau_2$.  That is, $\tau_2$ is finer.
