The category of 𝐷-completely regular spaces is simple

In a recent paper H. Brandenburg characterized the objects of the epireflective hull of all developable spaces-that are those spaces which are homeomorphic to a subspace of a product of developable spaces-by intrinsic properties. It is shown here that these spaces, called D-completely regular, can be generated from a single second countable developable space D which has the same cardinality as the reals. As an application of this result we obtain a new characterization of D-normal spaces analogous to Urysohn’s lemma and a new (external) characterization of perfect spaces (meaning every closed set is a G δ {G_\delta } ).</p

Developability and Some New Regularity Axioms

In a recent publication H. Brandenburg [5] introduced D-completely regular topological spaces as a natural extension of completely regular (not necessarily T1) spaces: Whereas every closed subset A of a completely regular space X and every x ∈ X\A can be separated by a continuous function into a pseudometrizable space (namely into the unit interval), D-completely regular spaces admit such a separation into developable spaces. In analogy to the work of O. Frink [16], J. M. Aarts and J. de Groot [19] and others ([38], [46]), Brandenburg derived a base characterization of D-completely regular spaces, which gives rise in a natural way to two new regularity conditions, D-regularity and weak regularity.</jats:p