2 T. A . CHAPMA N The interior of Q is where Ij is the open interval (—1,1), and B(Q) = Q - s is the boundary of Q. The reader should be aware of some very notable differences between the spaces Q, s and B(Q), and their finite-dimensional counterparts. For example B(Q) is dense in Q and s is not o-cornpact! For each n 1 we let I n = I-j x * x I n (the standardn-cell)i, Q n = I n x I n + 1 x (a Hilbert cube) , thus giving us a factorization Q = I n x Q n +i In general we always use 0 to represent (0,0,... ) e Qn . 3. Z-Sets. We now introduce R. D. Anderson's notion of a Z-set, which is certainly one of the single most important concepts in infinite-dimensional topology. A closed set A in a space X is said to be a Z-set in X provided that there exist arbitrarily small maps of X into X A. This means that for every open cover U of X there exists a map of X into X —A which is Dclose to the identity. Some examples of Z-sets are (i) the subset A of Q, where A C s is compact, (ii) any finite point-set in Q, (iii) the subset X x {0} of X x [0,1], arbitrary X, (iv) any closed subset of the combinatorial boundary 3M of a topological n-manifold M. (In connection with (iv) we note that no non-empty closed subset of the combinatorial interior of M can be a Z-set In M.) An embedding f: X -* Y is a Zembedding if f(X) is a Z-set in Y. In the following result we give some of the basic properties of Z-sets. In each assertion X is locally compact. 3.1. Theorem. (1) If A C X is a Z-set and h: X -* Y is a homeomorphismt then h(A) C Y is a Z-set (2) If A C X is a Z-set and B C A is closed, then B C X is a Z-set. (3) If A C X is closed and A = U°° , A n , where A n C X is a Z-set, then A C X is a Z-set n-i n M (4) If A C U C X, where A C X is closed, U is open and A C U is a Z-set, then A C X is a Z-set
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