The Collins–Roscoe structuring mechanism, D-spaces and related topics

We study the relationship between the Collins–Roscoe structuring mechanism and D-spaces. We introduce the notion of κ-sheltering and σ-sheltering point networks. We show that σ-sheltering (F) spaces are D  -spaces which improves the results of Gruenhage and Yuming, and also show that Σ⁎Σ⁎-products of σ-sheltering (F) spaces are also σ  -sheltering spaces. In addition, we show that ω1ω1-sheltering (F) spaces are DσDσ-spaces and transitively D  -spaces. As a corollary, monotonically normal ω1ω1-sheltering (F) spaces are paracompact.