Limit-like predictability for discontinuous functions
by
Christopher S. Hardin
Smith College
Coauthors: Alan D. Taylor (Union College)

Our starting point is the following question: To what extent is a function's value at a point x of a topological space determined by its values in an arbitrarily small (deleted) neighborhood of x? For continuous functions, the answer is typically "always" and the method of prediction of f(x) is just the limit operator. We generalize this to the case of an arbitrary function mapping a topological space to an arbitrary set. We show that the best one can ever hope to do is to predict correctly except on a scattered set. Moreover, we generalize the mu-strategy from [1] to obtain a predictor whose error set, in T₀ spaces, is always scattered.

The techniques are carried out in structures, which we call proximity schemes, that generalize both topological spaces and binary relations. A proximity scheme associates each point in a set X with a filter on X. To describe a topological space X, we associate each point with the filter generated by its neighborhoods. To describe a binary relation R on X, we associate each point x with the principal filter generated by the set of y such that xRy.

References:

[1] Christopher S. Hardin and Alan D. Taylor. A peculiar connection between the axiom of choice and predicting the future. American Mathematical Monthly, 115(2):91-96, February 2008.

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Date received: June 5, 2008