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Covering dimension in finite-dimensional metric spaces
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hall, Japheth |
| Copyright Year | 1973 |
| Abstract | Let P:2v->-2v be a structure in a topological space V such that P(0 )=0 , P({x})={x] if x e V, and P(Z) is closed if Zç V. If G is a covering of V, let Gz={Xe G:xe X}. If A' is a set and y is a set, let [X\ denote the cardinal number of X and X— Y= {x€X:x(fc Y}. Let n be an integer such that n^. — 1. dimP V is defined as follows: dinv K= -1 if V=0 . If Vjt 0 , then dim,, V= n if (1) for each finite open covering G of V, there is an open refinement Hof G such that \HX\ ̂/i +1 if x e V; and (2) there is a finite open covering G of V such that if H is an open refinement of G, then \HX\ ̂« + 1 for some x e V. We say that P has property (*) if for each nonempty open T<= V and each X^ V such that P(X) ?¿ V and x $ P(X-{¿-»whenever*e JTandeachxe[V-P(X)], [ Y-P{X)]r\ P(Xkj{x})^ 0. Theorem. If V is a metric space, P has property (•), B£ V, B is finite,P(B)= Vandx $ P(B-{x}) ifxeB, then dinv K= 1*1-1. |
| Starting Page | 274 |
| Ending Page | 277 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1973-0322828-2 |
| Volume Number | 41 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1973-041-01/S0002-9939-1973-0322828-2/S0002-9939-1973-0322828-2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1973-0322828-2 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
