Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements.

Can we give any description of $m$ as it relates to $n$?

Obviously $2\le m\le 2^n$ and the extreme cases are attained: $m=2$ for the indiscrete topology and $m=2^n$ for the discrete topology.

For $n=2$, $m=2, 3, 4$ are all possible.

For $n=3$, $m=2, 3,\dotsc, 6, 8$ are possible, only $m=7$ isn't possible.

For $n=4$, $m=2, 3,\dotsc, 10, 12, 16$ are possible, $m=11, 13, 15$ aren't possible.

Is it possible to obtain a result similar to Lagrange's theorem in group theory (in a finite group, the order of a subgroup must divide the order of the group)?

Obtainable sizes of topologies on finite sets, published as doi.org/10.1016/j.jcta.2009.05.002 $\endgroup$