Universality of the μ-predictor

Abstract

For suitable topological spaces X and Y , given a continuous function f : X → Y and a point x ∈ X, one can determine the value of f (x) from the values of f on a deleted neighborhood of x by taking the limit of f . If f is not required to be continuous, it is impossible to determine f (x) from this information (provided |Y | ≥ 2), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing f (x), called the µ-predictor, that will be correct except on a small set; specifically, if X is T0, then the guesses will be correct except on a scattered set. In this paper, we show that, when X is T0, every predictor that performs this well is a special case of the µ-predictor.