From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Cluster points and convergent subsequences
Date: 25 Apr 1999 15:42:02 -0500
Newsgroups: sci.math
In article ,
Erland Gadde wrote:
>Let X be a topological space. Let {x_n} be a sequence in X,
>and let x in X. We say that x is a _cluster point_ of {x_n}
>if every neighbourhood of x contains x_n for infinitely many n.
>We say that {x_n} _converges_ to x if every neighbourhood of x
>contains x_n for all but finitely many n.
>Now, I wonder if the following holds for arbitrary sequences
>and points in arbitrary topological spaces X:
> x is a cluster point of {x_n} if and only if {x_n} has a
> subsequence which converges to x.
>Now, it is clear that this is true if X is first countable
>(i.e. if there is a countable base around every point in X),
>and it is easy to see that the "if"-part is true in an arbitrary
>topological space. But what about the "only if"-part for an
>arbitrary topological space? Can someone please supply either
>a proof or a counterexample!
>Thanks in advance!
This is false in general. One of the simplest examples
is to take A to be the set of all sequences of 0's and 1's,
and X is the product of the two-element space {0,1} over A,
with the usual topology. We define the sequence x_n
by x_n(a) = a_n.
Since X is compact, this sequence has cluster points.
But no subsequence can converge to u, as convergence means
that, for each a in A, the a-th coordinate must be equal
to that of u for all a. But for any subsequence of the
x's, and any sequence q of 0's and 1's, there are a's for
which x_{n_j}(a) = u_j, so x_{n_j} cannot converge.
A subnet of a sequence need not be a subsequence.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558