ON EMBEDDINGS OF COMPACTA IN EUCLIDEAN SPACE = [0,1],
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ON EMBEDDINGS OF COMPACTA IN EUCLIDEAN SPACE J. L. BRYANT1
In [5] it was shown that any two embeddings of a fc-dimensional compact absolute neighborhood retract into euclidean w-space, En(n^5 and 2k+ 2 ^n), are equivalent by an ambient isotopy of E", provided the complement of each embedding is uniformly locally 1-connected (abbreviated 1-ULC). The purpose of this paper is to prove a generalization of the result stated above for embeddings of arbitrary
Theorem
E" (»^6
compacta
in En. The following
theorem
is the main result.
1. Suppose that X0 and Xx are k-dimensional
and 2k + 2^n)
such that En-Xt
compacta in
is 1-ULC for i = 0, 1, and
f: Xo—*Xx is a homeomorphism with d(x,f(x)) n) such that En-X
is 1-ULC,
map with d(x, f(x)) 0
each xEX.
in En
andf: X->£" is a there exists
an e-push h of (En, X) such that d(h(x), f(x)) En and e >0 are given as in the hypothesis of Lemma 2. Given 5>0, choose ??>0 such that if x, yEX and d(x, y)
