Some nonmetric, first countable, cancellative topological semigroups that are generalized metric spaces

AbstractA first countable topological group is metrizable. Certain nonfirst countable topological groups are “generalized metric spaces”. To elucidate the latter, we briefly list some theorems and examples. To indicate the difficulty of generalizing those theorems (especially the first countable case) even to “nice” topological semigroups we exhibit some nonmetrizable first countable cancellative topological semigroups that are, respectively: a quasi-metrizable, locally compact, Moore space; a nonquasi-metrizable Moore space; a Nagata space having no semimetric that is separately continuous in one variable; and a paracompact, quasi-metrizable, quasi-developable space that is neither developable nor stratifiable