Equicontinuous commutative semigroups of onto functions
Lawrence S. Husch · Czechoslovak Mathematical Journal · 1973
Equicontinuous (or regular) groups of transformations of a space onto itself have been studied extensively [1], [2], [5].In this note we investigate equicontinuous commutative semigroups G of functions of a space X into itself.We define a product on orbit closures which makes each orbit closure a commutative semigroup.This generahzes a result of D. MONTGOMERY on equicontinuous transformation groups [5].If X is compact Hausdorff and each ^ e G is onto, then each ^^ e G is a homeomorphism and each orbit closure is a topological group.This generalizes work of P. F. DuvALL, JR. and L. S. HUSCH [1] who considered the case when X is compact metric and G is generated by a single function.Finally it is shown that if X is compact Hausdorff then the closure of G in the space of continuous maps of X into itself with the compact-open topology is a topological group and each orbit closure is the continuous homomorphic image of G.