LECTURES ON HILBERT CUB E MANIFOLD S

3

(5) If A C X is a Z-set, X is an A N R and U C X is open, then A n U C U /$ a Z-$ef .

Proof . (1) an d (2) are trivial . Fo r (3) le t U be an ope n cove r of X an d for eac h n le t

fn: X -» X - A

n

be a map . It is possibl e to selec t the f

n

sufficientl y clos e to id so tha t the

infinit e lef t compositio n

f = r ^ f n f n - r * ' f 2 f 1

define s a ma p of X into X — A whic h is limite d by U. Sinc e X ca n be assigne d a complet e

metri c ther e is no proble m in choosin g the f

n

so tha t this limi t define s a map , i.e . is continuous .

(Jus t choos e the f

n

so tha t d(f

n

»id) 1/2

n

.) Als o ther e is no proble m in choosin g the f

n

so

tha t f is limite d by U. T o achiev e the requiremen t f(X O A = $ we observ e tha t by 4.1 belo w

we ca n choos e f ^ so tha t f-j (X ) is closed . Thu s f2,f3# • • • c a n be chose n so clos e to id tha t

f(X ) O A j = f . Inductivel y repeatin g this sam e ide a we ca n choos e the f

n

so tha t f{X ) O A

n

= $ ,

for al l n.

Fo r (4) we not e tha t f:U-* U —A ma y be chose n sufficientl y clos e to id y so tha t f extend s

by the identit y to a ma p f: X -» X — A . Thi s require s tha t f be chose n U-clos e to id y , wher e U

is an ope n cove r of U whic h consist s of ope n set s whos e diameter s approac h zer o as the y ge t

close r to X — U .

T o prov e (5) first observ e tha t by (3) we ma y assum e A compac t in U . Choos e ope n set s

U-j an d U 2 in X so tha t

A C U | C C l ^ l ^ ) C U

2

C CI

X

IU

2

) C U .

If f: X - X - A is chose n sufficientl y clos e to i d

x

, the n f(O

x

(U

2

) - U| ) is a subse t Off U - A

an d the restrictio n

f | CI

X

(U

2

) - U-, : CI

X

(U

2

) - U-, - U - A

is nomotopi c to the inclusion , with the entir e homotop y takin g plac e in U — A . (Her e we invok e

the fac t tha t "close " map s into an AN R (= U — A ) are homotopi c via a "small " homotopy. )

Usin g this homotop y we ca n easil y buil d a ma p

g : C l

x

( U

2

) - U

1

- ^ U - A

suc h tha t g|Bd

x

(U

2

) = id an d g = f on Bd

x

(U-|) . Exten d g to a ma p g : U - * U — A by definin g