J. C. Oxtoby obtains
the essence of the classical Baire category theorems in his pseudocomplete
spaces. In doing so, he reduces the role of points in his auxilary definitions of
quasiregular spaces and pseudobases. With a natural modification of his
definition of pseudobase that continues this reduction, a topological space is
quasiregular if it has a pseudobase of closed sets. With this change, strong
nesting is explicitly required in the definition of pseudocomplete spaces.
The change also leads to an equivalence relation on the topologies of a set
X: Topologies σ and τ are S-related if τ∗= τ ∖{ϕ} is a pseudobase for
σ. It also leads to conditions for which a topology finer or coarser than a
pseudocomplete topology is itself pseudocomplete. Several examples illustrate the
utility of quasiregularity, and there is a discussion of extensions of topological
spaces. In particular, it is noted that a T1-space is quasiregular iff it has a
quasiregular compactification, and a topological space has a quasiregular
one-point compactification iff the space has pseudobase of closed compact
sets.
