## Unique limits of sequences plus what implies Hausdorff?

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It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff.

What I am wondering is, if there is a (somehow weak) condition which one should add to "unique limits of sequences" to obtain a Hausdorff space. Would, for example, some countability help?

Somehow in the same direction: What is the central property which is needed for a space such that it can be non-Hausdorff but has unique sequence limits? Is there a whole class of non-Hausdorff spaces which admit unique limits for convergent sequence?

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The link in your question doesn't work for me. Anyway, it's easy to produce non-Hausdorff spaces with unique limits. The Frechet Topology has unique limits and is anti-Hausdorff (all open sets intersect). Another example is the co-countable topology on $\mathbb{R}$. See en.wikipedia.org/wiki/Fr%C3%A9chet_space and en.wikipedia.org/wiki/Cocountable_topology – David White Sep 7 2011 at 12:45
I believe having unique limits implies the space is $T_1$, so perhaps the question boils down to $T_1$ plus what implies $T_2$. – David White Sep 7 2011 at 12:50
I'm a bit confused. The Wikipedia page says that Frechet spaces are indeed Hausdorff. Also: Why do you have unique limits of sequences  in the cocountable topology? – Dirk Sep 7 2011 at 13:46
It seems you're right about Frechet spaces. I admit that I was just quoting a line from a topology course I took some time ago, and I didn't stop to question whether or not it's true. However, the cocountable one is right and I'll post the details as an answer – David White Sep 7 2011 at 15:23
Looking back at my notes, it wasn't Frechet spaces, but rather the Frechet topology on any space $X$, where you define $A$ to be closed iff $A$ is the set of limits of sequences in $A$. This topology can be used to construct an example of an anti-Hausdorff space with unique limits, but the answer below is just as good. – David White Sep 7 2011 at 15:31

## 4 Answers

First countable is enough. Let $x\neq y$ be two points in your space that cannot be separated by neighborhoods. Let $O_1,O_2,\ldots$ form a neighborhood base of $x$ and let $U_1,U_2,\ldots$ form a neighborhood base for $y$. Choose a sequence $(z_n)$ such that $z_n\in O_n\cap U_n$ for all $n$. Now $(z_n)$ converges to both $x$ and $y$.

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Thanks! To clarify: First countable + non-Hausdorff implies "non-unique limits" and hence, "unique limits" + first countable implies Hausdorff, right? – Dirk Sep 7 2011 at 10:45

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First countable is a nice answer. I don't think you're going to do better because you kind of need at least this assumption to say anything about sequences – David White Sep 7 2011 at 12:50

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Here's an example of a space which is not Hausdorff but which has unique limits...

Let $X = \mathbb{R}$ with the cocountable topology, i.e. a set is open iff its complement is countable. Clearly any two open sets intersect, because $\mathbb{R}$ is uncountable. So $X$ is non-Hausdorff. Now, suppose $(x_n)$ is a sequence which converges to $x$. Then $C =$ {$x_n\;|\;x_n\neq x$} is closed because it's countable. So $X-C$ is a neighborhood of $x$ and this means there is some $N$ such that for all $n>N$ $x_n\in X-C$, i.e. $x_n=x$ for large $n$. This means if $x_n\rightarrow y$ then $y=x$, proving limits are unique.

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This past weekend, entirely by chance, I came across a published paper that used the term "US-space" for the class of topological spaces having the property that no sequence can converge to more than one point. The google search just below seems to bring up some things that might be of use to you:

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The term "US-space" appears to go back to Albert Wilansky (1967) - ams.org/mathscinet-getitem?mr=208557 – François G. Dorais♦ Sep 12 2011 at 17:49

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Thanks! Especially the topospaces.subwiki.org is pretty cool... – Dirk Sep 12 2011 at 18:52

Here is an answer to Dirk's last question, Is there a class of non_Hausdorff spaces in which convergent sequences have unique limits''?

Yes. The so called KC-spaces or maximal compact spaces. These are spaces such that every compact subspace is closed.

(The 1967 Monthly article of Wilansky `Between T1 and T2' subsumes, references, or implies all of the following).

In a KC-space, convergent sequences have unique limits.

(Suppose xn-->x in the KC space X. The set {x,x1,x2,..} is compact and hence closed. Thus if y is not in the set {x,x1,x2,..} then the open set X minus {x,x1,x2,..} shows it is false that xn-->y. Thus if xn-->y then y=x or y=xn for some n. If y=xn for infinitely many indices n then y=x (since every KC space is T1 (since singletons are compact) and since constant sequences have unique limits in a T1 space). If y=xn for finitely many indices then (deleting y from the sequence x1,x,2...) we are left with a subsequence zn-->x, the knowledge that y is not zn, and the knowledge that y is in the set {x,z1,z2,...} and we conclude y=x).

To exhibit a large class of non-Hausdorff KC spaces let X be a non-locally-compact metric space ( for example the rationals) and let Y=X U {y} denote the Alexandroff compactification of X ( i.e. V is open in Y if V is open in X or if Y\V is a compact subspace of X).

The space Y is a KC space but Y is not Hausdorff.

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