The open sets in a topological
space are those sets A for which A0 ⊃ A. Sets for which A0−0 ⊃ A -“α-sets”- or
A0−⊃ A -“β-sets”- may naturally be considered as more or less “nearly open”. In this
paper the structure of these sets and classes of sets are investigated, and some
applications are given.
Topologies determining the same class of α-sets also determine the same class of
β-sets, and vice versa. The class of β-sets forms a topology if and only if the original
topology is extremally disconnected. The class of α-sets always forms a topology, and
topologies generated in this way-“α-topolgies”- are exactly those where all nowhere
dense sets are closed.
The class of all topologies which determine the same α-sets is convex in the
ordering by inclusion, the α-topology being its finest member. Most topologies
ordinary met with are the coarsest members of their corresponding classes; in
particular this is the case for all regular topologies.
All topologies determining the same α-sets also determine the same continuous
mappings into arbitrary regular spaces.