Effectively open real functions

Abstract

A function f is continuous iff the pre-image f(-1) vertical bar V vertical bar of any open set V is open again. Dual to this topological property, f is called open iff the image f vertical bar U vertical bar of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness.

By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping V (bar right arrow) f(-1) vertical bar V vertical bar being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f(-1)vertical bar V vertical bar. Analogously, effective openness requires the mapping U (bar right arrow) f vertical bar U vertical bar on open real subsets to be effective.

The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open. (C) 2006 Elsevier Inc. All rights reserved.