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