$T_1$ spaces and sequential limits

Question

Munkres introduce the $T_1$ separation axiom and proves the following theorem:

Let $A \subset X$ where $X$ is $T_1$. Then $x$ is a limit point of $A$ iff every neighborhood of $x$ contains infinitely many points of $A$.

He does this in the context of discussing when sequences have unique limit points. He goes on to prove that if a space is Hausdorff then it does have a unique limit point. But he stops talking about $T_1$ spaces. This made me wonder:

Do $T_1$ spaces have unique sequential limits?
If they don't, what point is Munkres trying to make? -- That is, "$T_1$ spaces don't have unique limits but they have this other more general property XYZ that gets us part way there", where I'm failing to understand what XYZ is.

https://math.stackexchange.com/questions/901154/unique-limits-in-t1-spaces — K. Y, Oct 07 '18 at 22:19
Why does Munkres have to be making a point beyond just the statement of the theorem? — Eric Wofsey, Oct 08 '18 at 00:25

Paul Frost · Accepted Answer · 2018-10-08T08:25:51.517

A very similar question (but not a real duplicate of the present one) is this: A separation axiom equivalent to uniqueness of limits of sequences . This should answer your second question: Uniqueness of sequential limits is a property stronger than $T_1$ but weaker than Hausdorff, and moreover that in general spaces it is more adequate to consider nets instead of sequences.

The reference provided by K. Y gives a negative answer to your first question.

$T_1$ spaces and sequential limits

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