Totally bounded endomorphisms on a topological ring

Abstract

Let $X$ be a topological ring. In this paper, we consider thethree classes ($btb$-bounded, $tbtb$-bounded, and $tbb$-bounded)of endomorphisms defined on $X$ and denote these classes by$B_{btb}(X),B_{tbtb}(X)$, and $B_{tbb}(X)$, respectively. Weequip them with an appropriate topology and we find somesufficient conditions under which, each class of theseendomorphisms is complete.