Universality of the $\mu $-predictor

Christopher S. Hardin¹

¹ 1 New York Plaza, 33rd Floor New York, NY 10004, U.S.A.

Fundamenta Mathematicae, Tome 220 (2013) no. 3, pp. 227-241

Résumé

For suitable topological spaces $X$ and $Y$, given a continuous function $f:X\to Y$ and a point $x\in X$, 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 $|Y|\geq 2$), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing $f(x)$, called the $\mu $-predictor, that will be correct except on a small set; specifically, if $X$ is $T_0$, then the guesses will be correct except on a scattered set. In this paper, we show that, when $X$ is $T_0$, every predictor that performs this well is a special case of the $\mu $-predictor.

Détail

DOI : 10.4064/fm220-3-4

Keywords: suitable topological spaces given continuous function point determine value values deleted neighborhood taking limit required continuous impossible determine information provided geq author alan taylor showed there nevertheless means guessing called predictor correct except small set specifically guesses correct except scattered set paper every predictor performs special predictor