The category of 𝐷-completely regular spaces is simple

Abstract

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 <italic>D</italic> -completely regular, can be generated from a single second countable developable space <bold>D</bold> which has the same cardinality as the reals. As an application of this result we obtain a new characterization of <italic>D</italic> -normal spaces analogous to Urysohn’s lemma and a new (external) characterization of perfect spaces (meaning every closed set is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi> δ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ).