Fix Point Theorem for Compact Spaces

The articles [10], [11], [1], [5], [8], [7], [12], [3], [9], [4], [2], and [6] provide the notation and terminology for this paper. In this paper M denotes a non empty metric space. One can prove the following proposition (1) Let F be a set. Suppose F is finite and F 6= / 0 and F is ⊆-linear. Then there exists a set m such that m ∈ F and for every set C such that C ∈ F holds m⊆C. Let M be a non empty metric space. A function from the carrier of M into the carrier of M is said to be a contraction of M if: (Def. 1) There exists a real number L such that 0 < L and L < 1 and for all points x, y of M holds ρ(it(x), it(y))≤ L ·ρ(x,y). Next we state the proposition (2) Let f be a contraction of M. Suppose Mtop is compact. Then there exists a point c of M such that f (c) = c and for every point x of M such that f (x) = x holds x = c.