The fixed point property of an M-retract and its applications

The present paper studies both the fixed point property (FPP for brevity) and the almost fixed point property (AFPP for short) for Marcus-Wyse (M-, for short) topological spaces by using an M-retract. Under M-topology, let SNM(x) be the smallest open neighborhood of x. In this paper we prove that every M-connected subspace of SNM(x) has the FPP. Let SCM4 be a simple closed M-curve with four elements in Z2. Besides, consider an M-topological plane (X,γX) (see Definition 5) such that SNM(x)⊂(X,γX), where x is a double even point or an even point. Then, proving that SCM4 is an M-retract of (X,γX), we propose that every M-topological plane does not have the AFPP. Finally, the present paper proves the following: consider M-connected spaces (Xi,γXi), where Xi⊂Z, 2≤|Xi|≨∞, i∈{1,2}. Then the Cartesian product as an M-topological subspace (X1×X2,γX1×X2) does not have the AFPP.