Fixed point theorems for topological contractions and the Hutchinson operator

• Main Authors: Morayne, Michał, Rałowski, Robert
• Format: Journal Article
• Language: English
• Published: 04-08-2023
• Subjects: Mathematics - General Topology
• DOI: 10.48550/arxiv.2308.02717

For a topological space $X$ a topological contraction on $X$ is a closed
mapping $f:X\to X$ such that for every open cover of $X$ there is a positive
integer $n$ such that the image of the space $X$ via the $n$th iteration of $f$
is a subset of some element of the cover. Every topological contraction in a
compact $T_1$ space has a unique fixed point. As in the case of metric spaces
and the classical Banach fixed point theorem, this analogue of Banach's theorem
is true not only in compact but also in complete (here in the sense of
\v{C}ech) $T_1$ spaces. We introduce a notion of weak topological contraction
and in Hausdorff spaces we prove the existence of a unique fixed point for such
continuous and closed mappings without assuming completeness or compactness of
the space considered. These theorems are applied to prove existence of fixed
points for mappings on compact subsets of linear spaces with weak topologies
and for compact monoids. We also prove some fixed point results for $T_1$
locally Hausdorff spaces and, introduced here, peripherally Hausdorff spaces.
An iterated function system on a topological space, IFS, is a finite family of
closed mappings from the space into itself. It is contractive if for every open
cover of $X$ for some positive integer $n$ the image of $X$ via a composition
of $n$ mappings from the IFS is contained in an element of the cover. We show
that in $T_1$ compact topological spaces the Hutchinson operator of a
contractive IFS may not be closed as the mapping in the hyperspace of closed
subsets of the space. Nevertheless, the Hutchinson operator of a contractive
IFS has always a unique fixed point.