Test spaces for infinite sequential order

Abstract

The collection of test spaces for the ordinal invariant σ on the category of regular sequential topological spaces is determined. The spaces K α, α < w 1 defined by Archangel'skiǐ and Franklin are shown to have this property. To establish this a collection of spaces T and a topological extension for the spaces in T 1 the principal extension, are defined. In particular, if X is a regular sequential space and σ( X)= α< ω 1, then X contains a subspace T α ϵ T such that the principal extension of T α is K α. Furthermore, if X is a Hausdorff sequential space which contains a copy of K α, then σ( X)⩾ α. A variety of sequential closure formulas are derived and several examples are presented.