Formalizing a Continuous Function in Topological Spaces Using Coq

We present a formalization work in Coq about the continuous functions of topological spaces. The continuity of functions for topological spaces plays an essential role in general topology, which was the origin of rapid developments in general topology. Our formalization work about the continuous functions is based on the system of the axioms of Morse-Kelley, and this axiom system can provide a solid foundation for mathematics. The concepts of continuous function are then introduced and their equivalent definitions are investigated. We transpose the definitions and properties of continuous functions into a formal description using the Coq theorem prover, which demonstrated the rigorous and reliable of formal mathematics. This work shows that the definition of continuous function in topological spaces can also be described by various basic topological concepts. And our formalization work lays a foundation for the formalization of the theory of point-set topology.