Constructions and applications of rigid spaces-II

Kannan, V. ; Rajagopalan, M. (1978) Constructions and applications of rigid spaces-II American Journal of Mathematics, 100 (6). pp. 1139-1172. ISSN 0002-9327

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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.

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