Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Complexity theory of real functions
Complexity theory of real functions
Complexity and real computation
Complexity and real computation
Theoretical Computer Science - Special issue on computability and complexity in analysis
An effective Riemann mapping theorem
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
Foundation of a computable solid modelling
Theoretical Computer Science
Computability in linear algebra
Theoretical Computer Science
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
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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 imagef[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. 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.