Vol. 95, No. 1, 1981

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Quasiregular, pseudocomplete, and Baire spaces

Aaron R. Todd

Vol. 95 (1981), No. 1, 233–250

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.

Mathematical Subject Classification 2000
Primary: 54E52
Received: 12 October 1979
Published: 1 July 1981
Aaron R. Todd