Abstract
Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } .
Highlights
To make the paper self-contained, we recall that a mathematical object X has the FPP if every well-behaved mapping f from X to itself has a point x ∈ X such that f ( x ) = x
K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y \ { p}
Studied a certain fixed point Theorem in semimetric spaces and further, Reference [2] explored a coincidence point and common fixed point theorems in the product spaces of quasi-ordered metric spaces
Summary
Motivated by the Tarski-Davis theorem [8,9] on a lattice and Kuratowski’s question [10,11] on the product property of the FPP on a peano continuum (or a compact, connected and locally connected metric space), many works dealt with the FPP for ordered sets and topological spaces Some of these include References [3,10,12,13,14,15,16,17,18].
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