A further improvement of a minimax theorem of Borenshtein and Shul’man

If (X, τ) is a topological space, we denote by τs the topology on X whose members are the sets A ⊆ X such that X \A is sequentially τ -closed. Clearly, τs is stronger than τ . We note that a function f : X → R is sequentially τ -lower semicontinuous if and only if it is τs-lower semicontinuous. We also recall that a real function φ on a convex subset C of a vector space is said to be quasi-concave if, for every r ∈ R, the set {x ∈ C : φ(x) > r} is convex.