g-hcmotopy equivalences 7

the property that pg: Y - * Y is (proper) a-hcmotopic to Id^ and

gp: X - * X is (proper) p (a)-honotopic to Idx . By [13], a

proper map p: X - • Y between ANR's is a cell-like map if and only

if p is a proper a-hcmotopy equivalence for every open cover a

of Y .

A surjective map p: X - * Y is said to be an a-fibration if

for all maps F: Z x I - * y and FQ: Z - * X for which pF0 = F ,

there exists a map G: Z x I • * x such that GQ = F and pG is

a-close to F . If a proper map p: X - Y is an a-fibration for

every open cover a of Y , then we say that p is an approximate

fibration [9].

A surjective map p: X - * Y is said to be an a-lifting map if

for all maps F: Z x i • y and F: Z x dl - X for v^iich

pF = F|z x 9i , there exists a map G: Z x I - x such that

G|Z x 31 = F and pG is a-close to F . Suppose a and 3 are

open covers of Y and a star refines ( 3 , then any a-hanotopy

equivalence p: X -* - Y is a g-lifting map.

Let p: X - Y be a map. An embedding g: Y • X is called

an a-cross section if pg is a-close to Id . We say that p

has approximate cross sections if p has an a-cross section for

every open cover a of Y .

All our embeddings are proper otibeddings.