All spaces are assumed to be locally compact separable metric
spaces unless otherwise stated. We use d to denote the distance
function regardless of the space. If X c w and a is an open
cover of W , then N (X) denotes the union of all elements in a
that intersect X . If e 0 , we may regard e as the open
cover obtained from all open balls of diameter e . Suppose X c w ,
then X denotes the closure of X in W . Given a continuous map
f: X + Y we define Sf= {x e xlf^f(x) f {x}} .
We let
denote Euclidean n-space, I the closed unit
interval [0,1] and Dn = [0,l]n cjRn . For 0 r 1/2 , define
D* =
. If x,y
, then d(x,y) = Hx-yll .
Let NT be an m-manifold. We let ft denote the interior of
M and 8M the boundary of M . We use the exponent only when we
wish to emphasize the dimension.
Let Q be a combinatorial manifold and K a subccmplex of Q ,
then N(K,Q") denotes the second derived neighborhood of K in Q
and 3N(K,QM) denotes the boundary of N(K,Q") .
Let X and Y be topological spaces and a an open cover of
Y .
We say that a (proper) surjective map p: X -* Y is a (proper)
a-homotopy equivalence if there exists a proper map g: Y X with
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