A non-Hausdorff space with unique limit

Question

Can I find a topological space $X$ such that every convergent sequence in $X$ has a unique limit in $X$, but $X$ is not Hausdorff?

Yes, if you look hard enough. Note that it's essential that you're considering only convergent sequences, not convergent filters or nets. — Daniel Fischer, May 12 '16 at 18:27

score 13 · Accepted Answer · answered May 12 '16 at 19:22

13

The co-countable topology on an uncountable set $X$ (like the reals) (the only closed sets are $X$ and all sets that are at most countable) is such a space.

The only convergent sequences are eventually constant and so the limits of convergent sequences are unique. But all non-empty open sets intersect, so $X$ is in fact anti-Hausdorff.

answered May 12 '16 at 19:22

Henno Brandsma

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We met again!! :) Thank you so much for replying – Silent May 12 '16 at 19:44

A non-Hausdorff space with unique limit

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