Metric spaces and enriched categories 2

[flengyel]

Lawvere generalized topological spaces, which arise as categories

enriched in the monoidal category |R| of nonnegative real numbers

with unit 0 and tensor +, are also known in the literature

as quasimetric spaces; see "All Topologies Come From Generalized Metrics"

Ralph Kopperman in The American Mathematical Monthly, Vol. 95, No. 2 (Feb., 1988), pp. 89-97

Kopperman shows that all topological spaces arise from quasimetric spaces, where the quasimetric has values in a "value semigroup."