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 , 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
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.