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