Rigidity of continuous quotients

We study countable saturation of the metric reduced products and introduce continuous fields of metric models 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 that the \vCech--Stone remainder of the half-line has only trivial automorphisms is independent from ZFC. The consistency of this statement follows from Proper Forcing Axiom and this is the first known example of a connected space with this property.