Base-paracompact spaces

Base-cover paracompactness

“…The irrationals P with the usual topology are not totally para(meta)compact [7], [1] (Konstantinov); see also [16]. Thus total paracompactness is very restrictive, which led John Porter to define and study base-base paracompactness and base paracompactness [23], [24]. His proof that metrizable spaces are base-base paracompact worked for what we called base-cover paracompact spaces [21], [22].…”

“…S, M and P are base-cover hypocompact (Theorem 1.8 here, and [22]); the countable sequential fan is totally paracompact. Base-base paracompact spaces are D-spaces, and Lindelöf spaces are base paracompact, but it is unknown if paracompactness implies base paracompactness, or if the latter implies base-base paracompactness [23], [24]. This is related to the following question of Eric van Douwen, versions of which have been discussed in [4], [6], [9], [10], [12], [26]: Is every Lindelöf space a D-space?…”

“…Taking the product with a compact factor easily destroys base-cover paracompactness, and in Section 3 we show that a paracompact space is locally compact if and only if its product with every compact space is base-cover paracompact. In Section 4 related known properties including total paracompactness [13], base-base paracompactness and base paracompactness [23], [24], and the D-space property [10] are discussed. Definition 0.1.…”

A note on base-paracompact and monotone base-covering properties

“…A topological space X is base-paracompact [13] (base-metacompact [9]) if there is a base B for X with |B| = w(X) such that every open cover of X has a locally finite (point-finite) refinement by members of B. In [13] and [9], some properties of base-paracompact spaces and base-metacompact spaces are investigated. In [5], it is proved that every paracompact generalized ordered topological space (ab.…”

“…A subspace M of a topological space X is called base-paracompact (base-metacompact) relative to X if there is a base B for X with |B| = w(X) such that for every family U of open subsets of X with M ⊂ U there is a subfamily r(U) of B which is locally finite (point-finite) in X such that r(U) ≺ U and M ⊂ r(U). The notion of base-paracompact relative to a space X is introduced in [13]. A subspace M of a topological space X is called monotonically base-paracompact (base-metacompact) relative to X if there is a base B for X with |B| = w(X) such that for every family U of open subsets of X with M ⊂ U there is a subfamily…”