Generalized Fan-Ha section theorem

In order to study the Fan Ha section theorem, H space and local intersection property are introduced. Relaxing convexity and closedness of some sets, Fan Ha section theorem and minimax theorem are generalized to H space, that is, let ({X{Γ A}), (Y,{Γ D}) be two Hausdorff H spaces, BCX×Y such as follows: a for each x∈X, {y∈Y, (x,y)B} is H convex or empty; b for each y∈Y, {x∈X, (x,y)∈C} is compactly closed in X; c for each x∈X, there exists a nonempty set A xX×Y, A x=P x×Q x, P x is a compactly closed subset in X, Q x is a compact subset of Y. d Further, suppose that there exists a nonempty compact subset K of X and for each finite subset N of X, there exists a compact subset L N of X containing N such that ① for each y∈Y, L N∩{x∈X, (x,y)∈A z for all z∈L N} is acyclic; ② for each x∈L N\K, {y∈Y,(x,y)∈A z for all z∈L N}{y∈Y, (x,y)∈B}; e for each x∈K, {y∈Y, (x,y)∈A z for all z∈X}=. Then there exists a point x 0∈X such that {x 0}×YC.