Shrinking bounded domains to totally bounded ones

Abstract

The Kuratowski measure of noncompactness or the measure of nondensifiability provide direct approach to topological fixed point theorems or to existence issues of generalized fractals. We point out that these measures are not so distinguished as they appear at first glance: Requiring quite simple properties on a set-function, we can prove analogous results. The method behind (the main result of this note) is a reducing principle which allows to shrink bounded and closed domains to compact ones. In the approach, the Knaster–Tarski and the Kantorovich Fixed Point Theorems play a key role.