Kuratowski monoids of \(n\)-topological spaces. (English) Zbl 1398.54003

The 14-set closure-complement theorem of C. Kuratowski [Fundam. Math. 3, 182–199 (1922; JFM 48.0210.04)] is generalized to a set \(X\) endowed with finitely many topologies, which are linearly ordered by inclusion. Following S. Plewik and M. Walczyńska [J. Math. 2013, Article ID 289854, 9 p. (2013; Zbl 1284.54006)], the authors estimate the sizes of the composition monoids generated by the operation of complement and the operation of closure for \(n\) pairwise comparable topologies \(\tau_1\subseteq \tau_2 \subseteq \cdots \subseteq \tau_n\) on a given set \(X\). The most important observation states that the considered monoids consist of irreducible words \(x_1\circ x_2 \circ \cdots \circ x_n\) which are alternating, i.e., every second term is an interior operator and the remaining terms are closure operators. If \(\tau_1\subseteq \tau_2 \subseteq \cdots \subseteq \tau_n\) is fulfilled, then the word problem is resolvable by the principle: If \(x_1\circ x_3 = x_1\) and \(x_2\circ x_4 = x_4\), then \(x_1\circ x_2 \circ x_3 \circ x_4=x_1\circ x_4\). The particular case of this rule works also for a single topology. Namely, if \(x_1(A) =x_3(A) = \text{interior}(A) = A^{c-c}\) and \(x_2(A) =x_4(A) = \text{closure}(A) = A^{-}\), then we get \(A^{c-c-c-c-} = A^{c-c-},\) which suffices to justify the 14-set closure-complement theorem of Kuratowski.


54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
20M20 Semigroups of transformations, relations, partitions, etc.
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