In this paper the Alexandroff one point compactification of the 2-dimensional
Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line)
is called the infinite K-sphere (resp. the infinite K-circle). The present
paper initially proves that the infinite K-circle has the fixed point
property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of
all continuous self-maps f of the infinite K-circle. Next, we address the
following query which remains open: Under what condition does the infinite
K-sphere have the FPP? Regarding this issue, we prove that the infinite
K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally,
we compare the FPP of the infinite K-sphere and that of the infinite
M-sphere, where the infinite M-sphere means the one point compactification
of the Marcus-Wyse topological plane.
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