I. Preliminarie s
Th e purpos e of this chapte r is to introduc e som e basi c terminolog y an d concept s whic h
will be neede d in the sequel . Additiona l materia l will be introduce d late r c i as it is needed .
1. Genera l Definitions . Fo r an y spac e X an d A C X , lnt
x
(A) , Bd
x
(A ) an d CI
X
(A ) wil l
denot e the topologica l interior , boundar y an d closure , respectively , of A in X . We use id ^
to denot e the identit y mappin g of X ont o itsel f an d the inclusio n of A into X wil l be
denote d by A C * X (or id ^ whe n the meanin g is clear) . A s usual , the subscrip t wil l be omitte d
whe n ther e is no ambiguity .
Al l mapping s (or maps ) are continuou s function s an d homeomorphism s will alway s be onto ,
while an embedding f: X -* Y is a homeomorphis m of X ont o f(X ) C Y . We use X = Y to
indicat e tha t X is homeomorphi c to Y . Ahomotopy F : X x I - * Y is a ma p (I =[0,1] ) an d
for t I we defin e F
t
: X -* Y by F
t
(x) = F(x,t) . Homotopi c map s will be denote d by f sr g.
If f,g:X-* Y are map s an d Y has a bounde d metri c d, the n we defin e
d(f,g ) = sup{d(f(x),g(x))|x€X} .
Fo r compac t spaces , d(f,g ) is quit e adequat e to describ e "closeness. " Fo i non-compac t space s
we use ope n cover s as follows : If f,g : X - Y are map s an d (J is an ope n cove r of Y , the n we
say tha t f is U-c/ose to g provide d tha t for eac h x E X , ther e exist s som e elemen t of U
containin g bot h f(x) an d g(x) . A ma p f: X -• Y is a near homeomorphism if for eac h ope n
cove r (J of Y ther e exist s a homeomorphis m of X ont o Y whic h is U-clos e to f.
Al l space s will be locall y compact , separabl e an d metri c (unles s otherwis e specified) . Whe n
ther e is no confusio n the lette r d will be use d to denot e the metri c of an y spac e unde r consideration .
2. Th e Hilber t Cube . Th e Hilbert cube Q will be represente d by the countabl e infinit e produc t
Q-iCi f i '
wher e eac h Ij is the close d interva l [-1,1] . It is well-know n tha t Q is a strongl y infinite -
dimensiona l compac t absolute retract (AR ) an d tha t ever y separabl e metri c spac e ca n be embedde d
in Q . Point s of Q wil l be denote d by q=(qj) , wher e q
{
I, , an d we use the metri c on Q
define d by
d((q
i
),(rj)) = S ~
1
| q
i
- r
i
| / 2 i .
1
http://dx.doi.org/10.1090/cbms/028/01
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