Effectively Open Real Functions

TL;DR

This paper explores the concept of effective openness in real functions, extending classical theorems to computable functions and revealing new classes of effectively open functions through logical and topological methods.

Contribution

It generalizes classical open mapping theorems to the computable setting, introducing effective openness and identifying new effectively open function classes.

Findings

Classical open mapping theorems can be effectivized for computable functions.

Several rich classes of functions are shown to be effectively open.

Effective openness can be characterized using Tarski's Quantifier Elimination.

Abstract

A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is open again. Dual to this topological property, f is called OPEN iff the IMAGE f[U] 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+->f^{-1}[V] being EFFECTIVE: Given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on open real subsets to be effective. By effectivizing classical Open Mapping Theorems as well as from application of Tarski's Quantifier Elimination, the present work reveals several rich classes of functions to be effectively open.