The topological structures of the spaces of diagonal and opposite diagonal functions with the uniform metric

Abstract

Diagonal and opposite diagonal functions play an important role in various applications of the copula theory. Denote the families of all 2-Lipschitz continuous functions from the unit closed interval to itself, diagonal functions and opposite diagonal functions, by L2(I), D and O, respectively. In this work, we study the topological structures of the sets D and O with the uniform metric d∞, and mainly show that there exist homeomorphisms H1,H2:(L2(I),d∞)→[0,1]×Q such that H1(D,d∞)=H2(O,d∞)={0}×Q, where Q=[−1,1]N is the Hilbert cube. This result allows us to deduce that the set of all (opposite) diagonal sections of (quasi-)copulas with the topology induced by the uniform metric is homeomorphic to the Hilbert cube Q, and hence has the fixed point property.