Douglas S. Bridges: Approaching topology constructively via apartness

In 2000, Luminita Vita and I began investigating axioms for a constructive theory of apartness between points and sets, and between sets and sets, as a possible constructive approach to topology. The culmination of this work came last October, with the publication of the monograph [3], in which we lay down what we believe to be the definitive axioms for (pre-)apartness and then develop the theory, with particular application to (quasi-)uniform spaces. Our development uses points, and is therefore in the style and spirit of Bishop's original constructive approach to analysis [1, 2]. As with analysis, so with topology: once the "right" axioms are used, the theory allows in a natural, if technically nontrivial, way (with one exception). In this talk I shall present the axioms for apartness and uniform spaces, and discuss various aspects of the resulting theory, paying particular attention to the connections between various types of continuity of functions.

References

[1] E. Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York 1967.

[2] E. Bishop and D.S. Bridges, Constructive Analysis, Grundlehren der Math. Wissenschaften 279, Springer Verlag, Heidelberg, 1985.

[3] D.S. Bridges and L. Vita, Apartness and Uniformity: A Constructive Development, in: CiE series "Theory and Applications of Computability", Springer Verlag, Heidelberg, 2011.