Box and nabla products that are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e19" altimg="si40.svg"><mml:mi>D</mml:mi></mml:math>-spaces

A space X is D if for every assignment, U, of an open neighborhood to each point x in there a closed discrete such that ⋃{U(x):x∈D}=X. The box product, □Xω, Xω with topology generated by all ∏nUn, where Un open. nabla ∇Xω, obtained from □Xω quotienting out mod-finite. weight X, w(X), the minimal size base, while d=cofωω. It shown are specific compact spaces and ∇Xω not D, but general: (1) hereditarily scattered either paracompact or finite height, metrizable (and w(X)≤d □Xω); (2) first countable w(X)≤ω1, consistently |X|≤c, w(X)≤ω1; (3) w(X)≤ω1.