Closed ideals in semigroups of continuous selfmaps
All topological spaces are assumed to be Hausdorff and \(S(X)\) is the semigroup of all continuous selfmaps of the topological space \(X\). The author provides \(S(X)\) with the topology of pointwise convergence. \(S(X)\) is not, in general, a topological semigroup with this topology. Indeed, if \(X\) is locally compact, then the compact-open topology is the smallest topology for which \(S(X)\) is a topological semigroup and the topology of pointwise convergence is generally smaller than the compact-open topology. Nevertheless, the topology of pointwise convergence still is, as the author points out, a natural topology for \(S(X)\) in that one does have a semitopological semigroup and this will be the topology on \(S(X)\) for the remainder of the discussion. The author shows that for certain spaces \(X\), the closure of the principal right ideal generated by \(f\in S(X)\) is \(\{g\in S(X): g[X]\subseteq\text{cl} f[X]\}\) and he shows that all principal left ideals are closed. He then goes on to describe those left and right ideals of \(S(I)\) (\(I\) is the closed unit interval) which are closed and he uses the previous results to show that the only closed two-sided ideals of \(S(X)\) are the ideal of all constant functions and \(S(X)\) itself. And now, some further comments are in order. First of all, Proposition 2.6 is false. It asserts that if \(X\) is completely regular and locally pathwise connected, then each map from a finite subset of \(X\) into \(X\) can be extended to a continuous selfmap of \(X\). In fact, it is not difficult to verify that if \(X\) is completely regular and contains an arc, then each map from a finite subset of \(X\) into \(X\) can be extended to a continuous selfmap of \(X\) if and only if \(X\) is pathwise connected. The error, however, is not at all a serious one. It just means that the class of spaces to which the other results apply is somewhat less extensive. As for the principal right ideals of \(S(X)\), one can show for any space \(X\) that \(f\circ S(X)\) is closed whenever \(f\) is a regular element of \(S(X)\) and there are a number of spaces \(X\) for which the converse holds. That is, \(f\circ S(X)\) is closed if and only if \(f\) is a regular element of \(S(X)\). One can verify, for example, that this holds if \(X\) is any local dendrite with finite branch number or any 0- dimensional metric space. Finally, in Theorem 3.1, the author describes the principal left ideals of \(S(X)\) for certain \(X\). Results of this nature have appeared in the literature previously. It follows from either Theorem (2.4) of [the reviewer and \textit{S. Subbiah}, Semigroup Forum 10, 283-314 (1975; Zbl 0299.54029)] or Theorem (2.3) of [\textit{B. B. Baird} and the reviewer, ibid. 22, 9-45 (1981; Zbl 0455.20050)] that if \(X\) is compact and has the extension property described in Theorem 3.1 of the paper under review, then \(g\in S(X)\circ f\) if and only if the decomposition of \(X\) induced by \(f\) refines that induced by \(g\). Corollary 3.2 readily follows from this and, in particular, it follows easily from this that every principal left ideal is closed.
scientific article; zbMATH DE number 35817