Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces

The weak, stable, and ordinary Lusternik–Schnirelmann categories of a finite $T_0$-space $X$ satisfy ${\mathrm{cat}}_w(X)\le {\mathrm{cat}}_s(X)\le \mathrm{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\ge 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\le a<b\le 2a$ and $c\ge b$, as well as every triple $(a,a,c)$ with $2\le a\le c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.