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It is a well-known fact that in Hausdorff spaces, limits of sequences are unique but the converse is not true. On the other hand, we know that the property of being Hausdorff is a separation axiom.

My question is that:

Is there any separation axiom equivalent to uniqueness of limits of sequences?

M.Ramana
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1 Answers1

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The concept of a net generalizes that of a sequence. See for example https://en.wikipedia.org/wiki/Net_(mathematics). It is well-known that a space $X$ is Hausdorff if and only if each net in $X$ has at most one limit (that is, limits of nets are unique). Consult a book on general topology, for example

Engelking, Ryszard. "General topology." (1989) - Proposition 1.6.7.

Sequences are in general not adequate to describe the topology of $X$. However, here is a relevant result (Engelking Proposition 1.6.17).

If each sequence in $X$ has at most one limit, then $X$ is a $T_1$-space. If in addition $X$ is first countable, then $X$ is Hausdorff.

You see that there exists a relationship to separation axioms, and we could define an axiom which says the limits of sequences are unique. This property is stronger than $T_1$ and weaker than Hausdorff, but to my knowledge it does not have an individual name. Perhaps somebody else can give a comment.

Paul Frost
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    Maybe we could call it $T_{1\frac {1}{2}}$. We could call a Uryssohn space ( distinct points have some nbhds with disjoint closures) a $T_{2\frac {1}{2} }$ space but nobody does. – DanielWainfleet Oct 06 '18 at 03:14
  • I just found https://math.stackexchange.com/q/467587. In this question spaces in which limits of sequences are unique are called "US spaces". I have never before heard about this. Via Google I found the definition in https://www.cmi.ac.in/~vipul/mathjourneys/contytopologysep.pdf, but it is remarked that "This is not a very standard term, and we are just using it for a bit of convenience." See also https://topospaces.subwiki.org/wiki/US_implies_T1. – Paul Frost Oct 22 '18 at 12:35
  • See also https://mathoverflow.net/q/74724. – Paul Frost Oct 22 '18 at 12:42