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I know that Hausdorff spaces ($T_{2}$ spaces) have uniqueness of the limits. Is there any weaker condition that implies it?

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  • @Arthur: There is at least one extensively studied property lying strictly between Hausdorffness and the desired property; see my answer. – Brian M. Scott Sep 09 '15 at 08:09
  • Also see http://math.stackexchange.com/questions/328725/if-every-compact-set-is-closed-then-is-the-space-hausdorff for an example of a space which is not Hausdorff but has unique limits. – pre-kidney Sep 09 '15 at 08:12
  • However in the presence of first-countability, the conditions are equivalent. – pre-kidney Sep 09 '15 at 08:13

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Spaces in which convergent sequences have unique limits are called $US$-spaces. Spaces in which compact sets are closed are called $KC$-spaces. All Hausdorff spaces are $KC$, all $KC$-spaces are $US$, and neither implication is reversible. In this answer I gave an example of a $US$-space that is not $KC$. This question contains another example. The Alexandroff extension (non-Hausdorff one-point compactification) $\Bbb Q^*$ of the rationals is a well-known example of a $KC$-space that is not Hausdorff.

The $KC$ property is the most natural one that I’ve seen that lies between Hausdorffness and the $US$ property.

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