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Let G be a monotone
decomposition of En, then G can be extended in a trivial way, to the monotone
decomposition G1 of En+1, where En = {(x1,⋯,xn,0) ∈ En+1}, by adding to G all
points of En+1 − En. If the decomposition space En∕G of G is homeomorphic to
En, En∕G is said to be obtained by a pseudo-isotopy if there exists a map
F : En × I → En × I, such that Ft(= F|En × t) is homeomorphism onto En × t,
for all 0 ≦ t < 1, F0 is the identity and F1 is equivalent to the projection
En → En∕G.
The purpose of this paper is to present a relation between these two notions. It
will then follow, that if G is the decomposition of E3 to points, circles and
figure-eights, due to R. H. Bing, for which E3∕G is homeomorphic to E3, then E4∕G1
is not homeomorphic to E4.
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