  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.
