A ghastly generalized $n$-manifold
Robert J. Daverman, John J. Walsh · Illinois Journal of Mathematics · 1981
valid for arbitrary metric spaces while the discussion of shrinkable defining sequences is limited to locally compact spaces. Defining sequences and decompositionsThe cell-like decompositions of manifolds mentioned in the literature, which often happen to be closed-0-dimensional ones (meaning that in the decomposi- tion space the closure of the image of the nondegenerate elements is 0- dimensional), frequently are described by defining sequences.In this section we generalize this standard notion of defining sequence.Although one can find more restrictive generalizations elsewhere, such as those given in [5] and [12], and for technical reasons aimed at producing specific kinds of decompositions one might wish to vary our definition somewhat, ours serves as an all- encompassing definition, because every upper semicontinuous decomposition of a locally compact metric space arises from such a defining sequence (Theorem 2.4).Let X be a space and ///a collection of subsets of X.For an arbitrary subset A of X define its star in /as st(A,/a) A ({ lVt A + }) and, recursively for any integer k >_ 1, define its kth-star in //as st(A, /)= st(st -'(A, g), g).When A {x}, x X, we write s?({x}, )simply as st(x, g).Now suppose X is a (locally) compact metric space.A defining sequence (in X) is a sequence 9 {, g, ...} satisfying the following axioms: AXIOM 1.For each the set ///i is a (locally) finite collection {M(i, 1), M(i, 2),..., M(i, r(i))} of compact subsets of X having nonempty, pairwise disjoint interiors.AXIOM 2. For each and each x e X, st3(x, 4i+ 1) Int st2(x, /i).The decomposition G of X associated with a definin9 sequence 5 {/{ 1, ' z, ...} is the relation prescribed by the rule: for any x X, G(x) is the subset of X consisting of all y X such that for every integer > O, y stZ(x, #i).Such a relation G on X obviously is reflexive and symmetric.The next lemma aids in showing that it is transitive.LEMMA 2.1.Suppose {(1, /2," "'} is a defining sequence on X.For each k > 0 and each x X, st(x, //// + ,) st3(x, #).