From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Topo Question Date: 8 Oct 1996 14:09:08 GMT In article , Phichet Chaoha wrote: >"Let A be a set and A' the set of accumulation points of A, > A and A' are disjoint ===> A is countable ?" Depends on whether this is for a topology class or an analysis class. In the former case, "set" might mean "subset of a topological space". In that case, the above implication is false. Take A' to be any (uncountable) set and A ={a}, a singleton. Give X = A'\union A the topology whose only open sets are the empty set, X, and {a}. In the latter case, "set" might mean "subset of the real line". In that setting, if A' and A are disjoint, then for each a in A there must be an e(a)>0 so that a is the only point of A in the interval (a - e(a), a + e(a) ). Adding up the lengths 2e(a) as appropriate, we see there can be only a countable number of a's in each bounded interval [-N, N], since an uncountable sum of positive terms must diverge. Taking the union of those subsets, we see A is a countable union of countable sets, hence countable. So it looks like you'll have to indicate the setting in which you want the above result to hold. dave