Constructions and applications of rigid spaces-II

Abstract

THEOREM A. Given an abstract group G, a subgroup H of G and a metric space X, one can construct a bigger metric space X* containing X as a closed subspace such that i. the homeomerphism group of X* is isomorphic to G, and ii. the isometry group of X* is isomorphic to H. THEOREM B. The extremally disconnects rigid spaces exist in such abundance that every topological space is a quotient of one such space. THEOREM C. Let X be any infinite set, and let f be any function from X into the set of all cardinal numbers not exceeding the cardinality of X. Then there is a connected (metrizable) topology on X such that for each x in X, the subspace X\{x} has exactly f(x) connected components. Besides proving these theorems, several methods of construting rigid extensions of spaces are discussed as tools for more important results.