Topologies of the quotient spaces induced by the M-topological plane and the infinite M-topological sphere

The present paper studies two topologies of the quotient spaces related to the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological plane. The topologies are exactly proved to be the cofinite particular point and the excluded two points topology. We also concern separation axioms and nowhere dense subset properties of the quotient spaces of the M-topological plane. Furthermore, we investigate both a particular point and an excluded point topological structure associated with the quotient spaces of the M-topological plane. Finally, we investigate the fixed point property of the compactification of the M-topological plane. Unlike the non-fixed point property of the Hausdorff compactification of the Euclidean topological plane, we prove that every continuous self-bijection of the compactification of the M-topological plane has a fixed point.