Rigidity of Continuous Quotients

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion “Stone–Čech remainder of the half-line has only trivial automorphisms” is independent from ZFC. Consistency of this statement follows from the Proper Forcing Axiom and this is the first known example of a connected space with this property. The present paper has two largely independent parts moving in two opposite directions. The first part (§§1–4) uses model theory of metric structures and it is concerned with the degree of saturation of various reduced products. The second part (§5) uses set-theoretic methods and it is mostly concerned with rigidity of Stone–Čech remainders of locally compact, Polish spaces. (A topological space is Polish if it is separable and completely metrizable.) The two parts are linked by the standard fact that saturated structures have many automorphisms (the continuous case of this fact is given in Theorem 3.1). By βX we denote the Stone–Čech compactification of X and by X we denote its remainder (also called corona), βX \ X . A continuous map Φ: X → Y ∗ is trivial if there are a compact subset K of X and a continuous map f : X\K → Y such that Φ = βf↾X, where βf : βX → βY is the unique continuous extension of f . Continuum Hypothesis (CH) implies that all Stone–Čech remainders of locally compact, zero dimensional, non-compact Polish spaces are homeomorphic. This is a Date: June 22, 2014.