B-7 - Cleavable (Splittable) Spaces

Abstract

The concept of cleavability generalizes the notion of one-to-one continuous map. Indeed, if ƒ : X →Z is a one-to-one continuous map from X into Z, then X is obviously cleavable over Z because the same map ƒ can be taken as fA for all subsets A of X simultaneously. It is therefore natural to expect that some topological results about spaces X that admit a one-to-one continuous map into a space from class P may remain valid under a weaker assumption that X is cleavable over P. Because a one-to-one continuous map from a compact space X onto a Hausdorff space is a homeomorphism, one may then expect that, for some ì naturalî subclasses P of the class of all Hausdorff spaces, a compact space cleavable over P must be homeomorphic to a subspace of some space from class P. A Čech-complete cleavable paracompact space is metrizable but there exists a nonmetrizable locally compact cleavable space. Nevertheless, a Čech-complete cleavable space always has a dense subspace metrizable by a complete metric. A locally compact metric space need not be cleavable. A cleavable complete metric space has cardinality 2ω .A complete metric space without isolated points is cleavable if and only if (iff) it has cardinality 2ω . A cleavable paracompact p-space is metrizable..