  In the paper \emph{K.~Sigmon} A strong homotopy axiom for Alexander
cohomology, Proc. Amer. Math. Soc. \textbf{31:1} (1972), 271--275 it was proven
that if $G$ is compact topological group or field then in the Homotopy Axiom
for Alexander-Spanier-Kolmogoroff cohomology the parameter segment $[0, 1]$ can
be replaced by any compact connected space $T$. The purpose of the paper is to
shows that the parameter space $T$ can not be replaced in general by locally
compact connected space.
