I want to find all topologies $\tau$ on finite set with $n$ elements such that all topologies have exactly one open single point or exactly $\{a_0 \}$ be in topology and cardinal of topologies $\tau$ be $2^{n-1} <|\tau |$.

Topologies on a finite set $X$ are in one-to-one correspondence with preorders on $X$.

I find one topology such that exactly has one open single point set  :

$$\tau=\{U:\{a_0\}\subseteq U\}\cup\{\varnothing\}. $$

which topologies admit exactly one open singleton and have more than half of the possible subsets open?How can I find all such topologies?

I want to find all topologies $\tau$ on finite set with $n$ elements such that all topologies have exactly one open single point or exactly $\{a_0 \}$ be in topology and cardinal of topologies $\tau$ be $2^{n-1} <|\tau |$.

Topologies on a finite set $X$ are in one-to-one correspondence with preorders on $X$.

I find one topology such that exactly has one open single point set:

$$\tau=\{U:\{a_0\}\subseteq U\}\cup\{\varnothing\}. $$

Which topologies admit exactly one open singleton and have more than half of the possible subsets open? How can I find all such topologies?