A Discrete Fixed Point Theorem of Eilenberg as a Particular Case of the Contraction Principle

Theorem 1.1 (Eilenberg). Let X be an abstract set and let (Rn)n=0 be a sequence of equivalence relations in X such that (i) X ×X = R0 ⊇ R1 ⊇ ··· ; (ii) ⋂∞ n=0Rn = ∆, the diagonal in X ×X ; (iii) given a sequence (xn)n=0 such that (xn,xn+1) ∈ Rn for all n ∈ N0, there is an x ∈ X such that (xn,x) ∈ Rn for all n∈ N0. If F is a self-map of X such that given n∈ N0 and x, y ∈ X , (x, y) ∈ Rn =⇒ (Fx,Fy) ∈ Rn+1, (1.1)