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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