Minimal invariant spaces in formal topology

Publisher： Cambridge University Press

E-ISSN： 1943-5886|62|3|689-698

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.62, Iss.3, 1997-09, pp. : 689-698

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Abstract

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).

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