By Marvin J. Greenberg

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**Example text**

2. X is an AR [resp. ANR] if and only if, for any space Y and closed subspace Z, every continuous map ~:Z ~ X extends to Y [resp. given ~, there is a neighborhood U of Z for which ~ extends to U]. 3. X is an ANR if and only if, for any space Y and decreasing sequence (Zn) of closed subspaces with Z = N Zn, every continuous ~:Z--~X extends to Z n for sufficiently large n. An AR is homeomorphic to a retract of the Hilbert cube I~, and is contractible; an ANR is homeomorphic to a (compact) retract of an open set in I=.

A shape theory for general (separable) C*-algebras, which exactly restricts to topological shape theory in the commutative case, was developed in [i]; this theory overcomes both drawbacks of [3]. It is hoped that this shape theory will play a role in noncommutative topology similar to that played by ordinary shape theory in the commutative case. 39 In this article, we give a survey of topological shape theory, the noncommutative theory of [i], and some applications, examples, and open problems.

I. It is well known that if a f u n c t i o n are zero there. are c h a r a c t e r i z e d INTRODUCTION that d i f f e r e n t i a l is zero on an open Peetre has shown by this p r o p e r t y operators set, are local, in the sense then also all its d e r i v a t i v e s that d i f f e r e n t i a l of locality: if operators K on is a linear ~n oper- 47 ator on the ort, such of f , space that then restriction with C= K support of K(f) i], C0(~) bra A = C0(X) support ([Peel; see also condition, conditions the G support somewhat In [BDR], this where EA Both [Bat for a the order for l i n e a r of C * - a l g e b r a s operators.