Generalization of the Comfort-Ross theorem. I
A subspace X of a topological space Y is said to be bounded in Y if every continuous, real-valued function on Y is bounded on X. Using the elegant observation that a necessary and sufficient condition for boundedness of X in Y is that only finitely many elements of any locally finite family of sets open in Y can intersect X, it is shown that the cartesian product of an arbitrary family of bounded subsets of topological groups is bounded in the product of the groups. [This, of course, generalizes the well-known theorem of Comfort and Ross.] It is also shown that a continuous pseudometric on a topological group is uniformly continuous on any bounded subspace, as is any real-valued continuous function defined on G. Furthermore, the universal, two-sided, left and right uniformities from the group are equal on X. If G is algebraically generated by X and X is bounded, then any real-valued function on X can be factored through a continuous homomorphism onto a group of countable weight. A union of countably many bounded subspaces is said to be sigma-bounded. A (Tychonoff) space X is then sigma-bounded iff all of the following are also sigma-bounded: the free topological group on X, the free topological Abelian group on X, and the free linear locally convex space on X.
scientific article; zbMATH DE number 4144285