By Bartolucci D., Pistoia A.
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B / 6 . I < b / 2 . ). I < b / 2 P . W The diagonal sequence (x;) is Cauchy. Returning to the proof of Theorem 6, we fix s E S and consider [x,,)where H,+l(x,) = s. We claim that [ U,] is a decreasing nest. To see this, if g , + l c W E Uu+2,we have s= Hfl+Z(X,+l)E Hn+Z(gn+l) c Hn+2(W)c Hn+l(U,+l), which implies that x, = H,L1l(s)E U,,+I,and g, C Vn+1 as well. Then by (a,),H,(U,) 3 Hn(Un+d,so V,, 3 U,+I for all n. ) that xn+k E un C N(g; en) C N(g; E ) , for some g E G. If (x,,(~)) is the Cauchy subsequence promised by Lemma 7, (Xn(i)) z E S and -+ p(z) = lim p(Xn(i)) = lim Hn(i)+l(xn(i)) = S.
H. Bing [2] set forth several conditions about countable cellular decompositions of E 3 implying shrinkability. His techniques depended on nothing intrinsically 42 11. The Shrinkability Criterion 3-dimensional ; the arguments functioned equally well in any Euclidean space. Among the first to recognize the potential generality of Bing’s methods was L. F. McAuley (21, who adapted them to nonmanifold settings by isolating a useful shrinkability property inherent in the notion of cellularity. With it we shall investigate conditions comparable to Bing’s implying shrinkability for decompositions of complete metric spaces.
This argument is probably more important than the result just established. Given an open cover V by sets with the favorite property of the moment and given any neighborhood W of g E G , we produced a homeomorphism h showing g C h-'( V ) C W , for some V E V. The consequence merits explicit statement. Proposition 12. Let 6 represent a topological property applicable to subsets of a given space. Suppose G is a shrinkable decomposition of a regular space S in which each point s E S has arbitrarily small neighborhoods satisfying 6.