Finite expansive homeomorphisms

We introduce the notions of metric finite expansive homeomorphism, orbit finite expansive homeomorphism, and topologically finite expansive homeomorphism on compact metric spaces, topological spaces, and uniform spaces, respectively. These notions coincide on compact metric spaces, but they are not the same on general metric spaces. We give some examples to show that the metric finite expansivity, orbit finite expansivity, and topological finite expansivity are weaker than the metric n-expansivity, orbit expansivity and topological expansivity, respectively. We state suitable conditions to imply that the metric finite expansivity is equal to the metric expansivity. We show that any regular recurrent point of a metric finite expansive homeomorphism f : X -> X is a periodic point and if X is an uncountable compact metric space, then Omega(f) is an infinite set. It is known that if there is an orbit expansive homeomorphism on X, then X is a T-1-space, in spite of it, we give an orbit finite expansive homeomorphism f : X -> X such that X is not a T-1-space. Then we show that if f : X -> X is an orbit finite expansive homeomorphism with the orbit finite expansive covering {U-i}(i=1)(n) , then X - U-i is an infinite set for all 1 <= i <= n. Finally we show that if f : X -> X is a homeomorphism on a compact Hausdorff space X, then f has a weak finite generator if and only if f is a topologically finite expansive homeomorphism. (C) 2018 Elsevier B.V. All rights reserved.