2

Let's say we have a finite set $S$ where $|S|=n$ and we take its powerset $X = \mathcal{P}(S)$ to generate a measurable space $(X, \mathcal{P}(X))$. From this we consider the probability space $(X, \mathcal{P}(X), \mathbb{P})$ where $\mathbb{P}$ is a uniform probability measure. What is the probability that a uniformly sampled $A \in \mathcal{P}(X)$ is a topology on S?

My intuition is that the probability will vanish as $n \rightarrow \infty$, but for finite sets it should be non-zero since there exists members of $\mathcal{P}(X)$ that are topologies on $S$.

Galen
  • 1,828
  • 2
    https://math.stackexchange.com/questions/3285831/estimates-on-number-of-topologies-on-a-finite-set says that the number of topologies on $[n]$ behaves approximately like $2^{n^2/4}$. – Nick Peterson Oct 12 '21 at 18:26
  • @NickPeterson That is very cool, and definitely applicable to the asymptotic analysis of the probability. Perhaps a useful approximation in topological data analysis, too. – Galen Oct 12 '21 at 18:27

0 Answers0