F–CONNECTED SPACES

Two new notions of connectedness and seperation in functional analysis is introduced. Let X be a real normed space, we say that a pair of open subsets A, B(⊆ X) is functionally seperation (briefly,F–separation) for C ⊆ X if there exists f ∈ X such that; f(A) ∩ f(B) = f(A) ∩ f(B) = ∅ and f(C) = f(A) ∪ f(B). The subset C of X is functionally connected (briefly, F–connected), if there is no any F–separation for C. We show that, in a Banach space X, the set C is F–convex if and only if it is F–connected. Moreover, we show that If {C α } is a chain of F–connected subsets of a norm linear space X, then C = α∈I is also F– connected. S α∈I C ∗ α