Some properties of the Sorgenfrey line and related spaces

Improving Negative Ion Beam Quality and Purity with a RF Quadrupole Cooler

“…A space X is a D-space [9], if for any neighborhood assignment ϕ for X there is a closed discrete subset D of X such that X = d∈D ϕ(d).…”

Base-cover paracompactness

“…This is possible, since each C i is discrete and hence with a closed-andopen union. See also [23], where it is shown that the Sorgenfrey line is base-base paracompact, and Remark 1.3 in [10], where it is shown that spaces satisfying somewhat stronger conditions than (a) and (b) above (the Sorgenfrey line among them) are ultraparacompact (i.e., every open cover has a disjoint open refinement). Remark 1.3.…”

“…It was shown in [10] that the irrationals as a subspace of the Sorgenfrey line S are not generalized left separated, although every F σ subspace of S is. In [5] the F σ subspaces of S were characterized as those subspaces that are continuous images of S. We define base-cover paracompactness and in Section 1 show that a subspace of S is F σ if and only if it is base-cover paracompact, if and only if it is generalized left separated.…”

“…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 new cardinal function on topological spaces

“…A number of classes of topological spaces can be characterized by means of neighbourhood assignments (for example, compact spaces, D-spaces, connected spaces, metrizable spaces) [14,3,5,9]. Neighbourhood assignments have also been employed to characterize special type of mappings, in particular those that are linked to Baire class one functions [1] [13].…”