Δ-related functions and generalized inverse limits

Abstract

For any continuous single-valued functions f, g : [0, 1] → [0, 1] we define upper semicontinuous set-valued functions F, G : [0, 1] ⊸ [0, 1] by their graphs as the unions of the diagonal ∆ and the graphs of setvalued inverses of f and g respectively. We introduce when two functions are ∆-related and show that if f and g are ∆-related, then the inverse limits lim −⊸ F and lim −⊸ G are homeomorphic. We also give conditions under which lim −⊸ G is a quotient space of lim −⊸ F .