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
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