Abstract: It is shown that if a "max-subadditive funtional" p(x) defined on some symmetric neighborhood Ū₀ of zero vector θ in a "b.f.-toplological group" X is "upper semi-cotinuous" at a point x₀ ∈ Ū₀, or "lower semi-continuous" in some neighborhood V(x₀)⊂Ū₀ and X is of second category; then p(x) can attain its supremum in Ū₀. And there is a similar conclusion for the γ-max-subadditive functional when its supremum is 0 and if Ū₀ is "pseudo-bounded" set in X.

Key words: γ-max-subadditive functional, Binary fissionable topological group, Pseudo bounded set