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