Tame topology is the name for the largely programmatic quest for a refoundation of topology and geometry that avoids ‘pathological’ objects like space-filling curves or counter-intuitive results like the Banach-Tarski paradox that occur in the traditional approach. The advantages of such a program that has among others been proposed by A. Grothendieck and W. Lawvere are philosophical by reconciling concepts and intuitions on space as well as technical by excluding pathological spaces.
The notion of o-minimal structure has been proposed as a reasonable candidate for an axiomatic approach to this hoped-for tame topology ( topologie modérée).
Michel Coste, An Introduction to O-Minimal Geometry , Lecture notes Pisa 1999. (pdf)
F. William Lawvere, Some Thoughts on the Future of Category Theory , pp.1-13 in Springer LNM vol. 1488 (1991).
F. W. Lawvere, Left and right adjoint operations on spaces and data types , Theoretical Computer Science 316 (2004) pp.105-111.
Lou van den Dries, Exponential rings, exponential polynomials and exponential functions , Pacific Journal of Mathematics 113 no.1 (1984) pp.51–66. (pdf)
Lou van den Dries, A Generalization of the Tarski-Seidenberg Theorem and Some Nondefinability Results , Bull. Am. Math. Soc. 15 (1986) pp.189-193. (pdf)
Lou van den Dries, Tame topology and O-minimal structures, London Math. Soc. Lecture Notes Series 248, Cambridge U. Press 1998.
Alexandre Grothendieck, Esquisse d’un Programme, section 5. English translation available in Geometric Galois Actions I (edited by L. Schneps and P. Lochak), LMS Lecture Notes Ser. 242, CUP 1997.
Yuliy Baryshnikov, Robert Ghrist, Euler integration over definable functions (pdf)
Jean Philippe Rolin,
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