Effectively open real functions
Copyright policy )Effectively open real functions
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.
added section on semi-algebraic functions; to appear in Proc. http://cca-net.de/cca2005
- University of Southern Denmark Denmark
- University of Paderborn Germany
Statistics and Probability, FOS: Computer and information sciences, computable analysis, Computer Science - Logic in Computer Science, Recursion theory, topology, Control and Optimization, Open mapping, Special maps on topological spaces (open, closed, perfect, etc.), Topology, Computable analysis, Theory of numerations, effectively presented structures, Numerical Analysis, Algebra and Number Theory, Quantifier elimination, Applied Mathematics, recursion theory, Quantifier elimination, model completeness, and related topics, open mapping theorem, Logic in Computer Science (cs.LO), quantifier elimination, F.4.1, Constructive real analysis, Constructive and recursive analysis
Statistics and Probability, FOS: Computer and information sciences, computable analysis, Computer Science - Logic in Computer Science, Recursion theory, topology, Control and Optimization, Open mapping, Special maps on topological spaces (open, closed, perfect, etc.), Topology, Computable analysis, Theory of numerations, effectively presented structures, Numerical Analysis, Algebra and Number Theory, Quantifier elimination, Applied Mathematics, recursion theory, Quantifier elimination, model completeness, and related topics, open mapping theorem, Logic in Computer Science (cs.LO), quantifier elimination, F.4.1, Constructive real analysis, Constructive and recursive analysis
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