In 1966 A. D. Wallace [6] presented the analogy between results present in the literature concerning periodicity properties of semigroups and of selfmaps of spaces. He asked whether gaps might be filled where there were theorems in one of these areas which still did not possess analogues in the other. More generally he asked, "Is there a mathematical system whose propositions properly subsume these analogous results?" The purpose of this paper is to present a system which does properly subsume some of the analogous situations which Wallace listed. In referring to the theorem which Wallace [6] listed, we use his numbers and symbols but to give the main idea we quote one such analogous pair of theorems in their entirety. Before doing so, we note that functions are continuous functions and semigroups are jointly continuous topological semigroups. Also, in the section that we quote, the spaces and semigroups are compact.
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