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Capturing Alexandroffness with an intuitionistic modality
by
Mamuka Jibladze
Razmadze Mathematical Institute, Tbilisi, Georgia
Considering finitary algebraic axiomatizations of the Cantor-Bendixson derivative operator d on subsets of a topological space, it is easy to characterize the spaces homeomorphic to preorders equipped with Alexandroff topology. Namely, they are precisely the spaces for which the operator d has a right adjoint.
The operator d restricts to closed sets; equivalently, the dual t of d restricts to the Heyting algebra of open sets. What amount of Alexandroffness can one recover by postulating existence of an adjoint for this restricted operator? It turns out that the class of spaces that one can capture in this way is strictly larger than that of Alexandroff spaces. Namely, one gets precisely those spaces in which the T0-reflection of the subspace of closed points of any open subspace is discrete.
Some illustrations of "Alexandroff-like behaviour" and "non-Alexandroff features" of such "pseudoalexandroff" spaces will be given.
Date received: May 9, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caug-35.