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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 $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Vill
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