I need to generate all finite topologies of orders up to $6$ (may be even $7$). The brute force approach applying the $3$ axioms of topology for all element combinations requires numerous iterations ($2^{2^n}$ for order $n$). The program I built works fine for orders up to $5$. Orders $6$ and beyond, it doesn't finish running (as it takes $2^{64}$ iterations for order $6$). I am curious to know if anyone has a better algorithm to solve this problem.
Thanks in advance!