In this paper,a FC-metric-space is introduced.

KKM type theorem in FC-space and its applications;

Coincidence Theorem and Applications in FC-space;

KKM Type Theorems and Coincidence Theorems in FC-spaces with Applications to Generalized Vector Equilibrium Problems;

Fixed point theorem and minimax theorems on FC-spaces;

Many topological spaces with abstract convexity structure are all FC-spaces.

This paper mainly further studies several focus problems of nonlinear functional analysis in finitely continuous topological spaces (in short, FC-spaces) without any convexity and linear structure, unites and generalizes some known results in recent literature.

Properties of FC-subspace generated by a subset and KKM type theorem on FC-spaces;

The author defines G-FC-spaces which are generalization of many general spaces such as FC-spaces and G-convex spaces and obtains a coincidence theorem for better admissible set-valued mapping defined on G-FC-spaces.