A SURVEY OF CLEAVABILITY

A space X is said to be cleavable over a class of spaces P if for every subset A of X there exist a space Y is-an-element-of P and a continuous mapping f : X --> Y such that f(A) and f(X - A) are disjoint and f(X) = Y. If X is cleavable over the class of all subspaces of a space Y, then Y is said to be cleavable over Y. Cleavability may be treated as a relativization of one-to-one continuous mappings. This paper is a survey which covers the first six years of the theory of cleavability. In particular, results on cleavability over the Hilbert space R(omega), the Euclidean spaces R(n), the real line R, the space Q of rational numbers and the space J of irrational numbers are presented.