Test spaces for infinite sequential order

AbstractThe collection of test spaces for the ordinal invariant σ on the category of regular sequential topological spaces is determined. The spaces Kα, α < w1 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 T1 the principal extension, are defined. In particular, if X is a regular sequential space and σ(X)= α