Limit-like predictability for discontinuous functions

Abstract

Abstract. 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 give a predictor whose error set, in T 0 spaces, is always scattered. Topological preliminariesAmong the topological spaces that we will be interested in are the ones arising from a partial ordering by either declaring a set to be open if it is closed upward in the ordering (the upward topology) or closed downward in the ordering (the downward topology). For example, the interval (−∞, 0] is open in the downward topology on the reals. These topologies are not typically T 1 , but they are always T 0 .We will be asserting that certain things happen except on a set that is "topologically small." On the real line R with the downward topology, we want these small sets to be the well-ordered subsets of R, and for the upward topology on an ordinal, we want these small sets to be the finite sets. The following well-known notions achieve both. Definition 1.1. A set S in a topological space X is weakly scattered if for every nonempty T ⊆ S there exists some x ∈ T and some neighborhood V of x such that V ∩ T is finite. We call such points weakly isolated points of T. The set S is scattered if the conclusion can be strengthened to V ∩ T = {x}, in which case these are called isolated points of T.What we are calling "weakly scattered" is called "separated" by Morgan [6]; he attributes the definition to Cantor. Every scattered set is weakly scattered, and it is straightforward to show that the two notions are equivalent in T 0 spaces. The concept of a weakly scattered set is a smallness notion in the sense that these sets are closed under the formation of subsets and finite unions.

Universality of the μ-predictor

“…We prove Theorem 1.3 at the end of this section, as it has not yet been published, but it is very similar to Theorem 2.4 of [HT09]; that result uses a slight modification of the µ-predictor to allow for a more economical proof (namely, it uses the µ * -predictor, which ignores finite differences, and which we visit in Section 5).…”

“…So it is surprising that scattered-error predictors exist, provided X is T 0 . The method of prediction introduced in [HT08] and generalized to topological spaces in [HT09] is the µ-predictor, which we now consider.…”

The mathematics of coordinated inference: a study of generalized hat problems

“…There is a game-theoretic characterization of weakly scattered sets in [HT09], special cases of which occured in [Fre90], that goes as follows. Given a space X and set S ⊆ X, Players I and II take turns, with Player I choosing elements of S and Player II choosing open sets.…”

“…In this section we consider modified versions of the µ-predictor such as the µ * -predictor (which ignores finite differences) from Chapter 7. One virtue of the µ * -predictor is that while the proof of Theorem 7.2.6 is about one page, the proof of the analogous result for the µ * -predictor is 11 lines [HT09]; that gives the µ * -predictor, perhaps, a greater claim to being the "right" approach. Another virtue of the µ * -predictor is that its willingness to overlook certain minor differences makes it work in some contexts where the µ-predictor can fail.…”

Prediction problems and ultrafilters on ω

“…Given a coloring g, agent x guesses correctly for g with P if S x (g) = g(x). For more background, see [HT08a], [HT08b], [HT09], and [HT10].…”