Convergent sequence and Hausdorff space

Question

It's trivial that any sequence in a Hausdorff space converges to at most one point. What about its inverse? E.g. If a topological space have the property that any sequence in it converges to at most one point, is it a Hausdorff space?

If this is not true, a counterexample will be appreciated.

Related : https://math.stackexchange.com/questions/2943399/a-separation-axiom-equivalent-to-uniqueness-of-limits-of-sequences — Arnaud D., Jul 15 '20 at 12:40

score 1 · Accepted Answer · answered Jul 15 '20 at 03:11

1

Let $X$ be an uncountable set and consider topological space $(X, \tau_{co-countable})$. Then every sequence converges to at most one point, but the space is not Hausdorff.in the cocountable topologyall convergent sequences are eventually constant, so they converge to at most one point.

answered Jul 15 '20 at 03:11

Unknown

3,105

Being eventually constant does not imply (in general ) that the limit is unique (e.g. the indiscrete topology), but it is enough here because $X$ is $T_1$. – Henno Brandsma Jul 15 '20 at 05:55
BTW, there is a (complicated) example of a sequential space that is not Hausdorff, but in which limits of convergent sequences are unique (a US space, as it's also called). – Henno Brandsma Jul 15 '20 at 05:58
@HennoBrandsma Yes! exactly. – Unknown Jul 15 '20 at 06:17
@HennoBrandsma What does US mean? – Unknown Jul 15 '20 at 06:18
Unique Sequential limits. – Henno Brandsma Jul 15 '20 at 09:00

Convergent sequence and Hausdorff space

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