发表机构
University of Santiago de Compostela; Shinshu University（圣地亚哥-德孔波斯特拉大学; 信州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造满足特定条件的有限T0空间，解决了Lusternik-Schnirelmann范畴弱、稳定与普通版本同时严格不等式的存在性问题，并实现了多种范畴三元组。
AI 中文摘要
有限$T_0$-空间$X$的弱、稳定和普通Lusternik--Schnirelmann范畴满足$\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$。我们给出一个一般构造，回答了Cárdenas、Flores、Quintero和Villar-Liñán提出的同时严格性问题。若$P$是弱可缩但不可缩的，且删除一个点后$P$变为可缩，则添加$m\geq 2$个不可比较的极大点可产生一个连通有限空间，其范畴三元组为$(1,2,m)$。此外，经过任意正数次重心细分后，其普通范畴等于$2$。将该构造应用于一个九点空间，可得到$m+9$个点上的例子，特别地，一个十二点例子使得两个不等式均为严格。利用在不相交并下的可加性，我们还实现了所有满足$1\leq a<b\leq 2a$且$c\geq b$的三元组$(a,b,c)$，以及所有满足$2\leq a\leq c$的三元组$(a,a,c)$。最后我们提出关于连通实现和展现同时严格性的有限空间最小基数的问题。
英文摘要
The weak, stable, and ordinary Lusternik--Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. 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 $m\geq 2$ 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 $1\leq a<b\leq 2a$ and $c\geq b$, as well as every triple $(a,a,c)$ with $2\leq a\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.
Comments15 pages, 3 figures. Comments are welcome
