From: Brandsma Subject: Re: Non-hausdorff spaces and convergent sequences Date: Mon, 11 Jan 1999 12:46:14 +0100 Newsgroups: sci.math Keywords: Failure of uniqueness of limits in non-Hausdorff spaces Martin van Hensbergen wrote: > Hello, > > It is easy to prove that in a Hausdorff space either a sequence > converges to exactly one point or no point at all. > > I suppose the converse isn't true, but I haven't been able to prove it. I > also thought about constructing an example of a space where each sequence > converges to exactly one point or no point at all, but is not Hausdorff. I > have failed in finding such a space. I've tried trivial and discrete > topologies and some others but I can't seem to construct one. > The converse is very non-true, indeed. For a recent refence (as the construction is too involved to repeat here, but in this paper there are references to earlier examples which are easier (but are less "extreme"); BTW your question is an old one, examples exist from around 1940 and earlier..): AUTHOR = {van Douwen, Eric K.}, TITLE = {An anti-{H}ausdorff {F}r\'echet space in which convergent sequences have unique limits}, JOURNAL = {Topology Appl.}, FJOURNAL = {Topology and its Applications}, VOLUME = {51}, YEAR = {1993}, NUMBER = {2}, PAGES = {147--158}, ISSN = {0166-8641}, CODEN = {TIAPD9}, Here a space is constructed in which every two non-empty open sets intersect (anti-Hausdorff), but in which every convergent sequence has a unique limit, and where the topology is Frechet, ie a set A is closed iff A equals the set of limits of sequences from A. (so the topology is determined by the sequences; this is certainly not true for general topological spaces). This last feature was in fact the reason for constructing the space, as others were known with weaker properties (but also anti-Hausdorff). Hope this answers your question, Henno Brandsma. > > Did I miss something? Could anyone give me a hint? My book remains silent > on the subject... > > Thanks in advance, > > -Martin ============================================================================== From: marcus00@supernet.ca (Francois G. Dorais) Subject: Re: Non-hausdorff spaces and convergent sequences Date: Wed, 13 Jan 1999 15:41:53 GMT Newsgroups: sci.math On Wed, 13 Jan 1999 10:31:50 +0100, Brandsma wrote: >Francois G. Dorais wrote: > >> In every non-Hausdorff space there is a sequence which converges to at >> least two points. The proof is easy: Suppose that E is a non-Hausdorff >> space, and let p and q be two distinct points of E such that every >> open neighborhood of p contains q (so p and q exemplify that E is >> non-Hausdorff). Let the sequence q(n) = q for all n. Then the sequence >> converges to both p and q. >> >> Francois >Well, this does not work: if you have a point p every neighbourhood of which >contains q your space is not T_1. So your proof shows the space must be T_1. >The space can be very non-Hausdorff indeed, as the example I mentioned in >another post shows. BTW non-Hausdorff means that there are distinct p and q >such that every neighbourhood of p and every neighbourhood of q intersect. It >does not mean that q is contained in every neighbourhood of p or vice versa.. >Henno Brandsma Oups! Sorry for the mistake. Let me make it up by answering the _real_ question. Fact 1. If X is a first countable non-Hausdorff space, then there is a sequence which converges to at least two points. Let x and y exemplify that X is non-Hausdorff and let and be neighborhood bases for x and y resp. WLOG the sequences and are decreasing with respect to inclusion. Since x and y exemplify that X is non-Hausdorff, for every n < omega we can pick a point z_n which lies in both Un and Vn. Then converges to both x and y. Fact 2. If X is a non-Hausdorff space, then there is a directed net which converges to at least two points. Recall that a directed net is a function z with range in X and domain a directed partial order D (a partial order in which any two elements have an upper bound). The directed net z is said to converge to a point x if for every neighborhood U of x we can find a d in D such that z(d') belongs to U whenever d' >= d. Thus said, let x and y exemplify that X is non-Hausdorff. Let M and N be neighborhood bases for x and y resp. Let D be the poset of all pairs (U, V) with U in M and V in N, and set (U, V) <= (U', V') iff U' is included in U and V' is included in V. Now for each pair (U, V) in D let z(U, V) be any point which U and V have in common. It is easily verified that z is a directed net which converges to both x and y. Even if we allow for sequences of lenght greater than omega, we cannot replace directed net by sequence in fact 2. It would be interesting to have an example of a non-Hausdorff space in which every sequence, whatever the lenght, converges to at most one point. If we restrict the definition of sequence to countable sequence, there is a trivial example of a non-Hausdorff space in which every countable sequence converges to at most one point: let X be an uncountable space and let the open sets of X consist of all co-countable subsets of X, then the relative topology of every countable subset of is discrete. Francois