A separation axiom equivalent to uniqueness of limits of sequences

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?

Solution 1:

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.