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