On the Extension of Seminormal Functors to the Category of Tychonoff Spaces

详细

Chigogidze proposed a construction of extending a normal functor from the category Comp to the category Tych. We can apply his scheme to seminormal functors and study the properties of the original functor which are preserved under extension. We introduce the concept of functor having an invariant extension from Comp to Tych because the very existence of this invariance plays a key role in the preservation of the properties of a seminormal functor in its extension. It is proved that the superextension functor λ has an invariant extension. We check that if a seminormal functor has an invariant extension then its extension preserves a point, the empty set, intersection and is a monomorphic functor. If this functor has finite degree then its extension is continuous and hence a seminormal functor in Tych. If the functor is of infinite degree then continuity may be lost. Namely, we show that the extension of λ for Tych is not continuous.