Background:
By a unique convergence (UC) topology, I mean a topology under which sequences converge to at most one point.
Suppose we are given a set $X$ and a topology $\mathcal T$ on $X$. It can be readily shown that each of the following implies the next:
(1) $\mathcal T$ is the discrete topology on $X$.
(2) $\mathcal T$ is a Hausdorff topology on $X$.
(3) $\mathcal T$ is a UC topology on $X$.
(4) $\mathcal T$ is a T$_1$ topology on $X$.
Moreover, (4) $\implies$ (3) if and only if $X$ is Dedekind-finite; (3) $\implies$ (1) if and only if $X$ is finite.
I've also shown the following:
Suppose that $X$ is an amorphous set, $\mathcal T$ a Hausdorff topology on $X$, $C:=\bigl\{x\in X:\{x\}\in\mathcal T\bigr\}$ the set of $\mathcal T$-clopen points of $X$. Then $C$ is cofinite in $X$, and putting $n=|X\setminus C|,$ the following are equivalent:
(a) $n\ne 0.$
(b) $\langle X,\mathcal T\rangle$ is compact; in particular, an $n$-point compactification of the discrete space on $C$.
(c) $\mathcal T$ is not the discrete topology on $X$.
From this, we see that for amorphous $X$, (2) need not imply (1)--so, unless there aren't any amorphous sets, it follows that D-finiteness need not be sufficient for (2) $\implies$ (1).
Finally, if $X$ is uncountable, then the cocountable topology on $X$ shows that (3) need not imply (2).
The Question: I intend to play more in the realm between (1) and (2) later (e.g.: regular Hausdorff, normal Hausdorff). At the moment, though, I'd like to refine what I have already as far as possible. Specifically, I'm curious about the following:
- Is it known to be true that (3) $\implies$ (2) if and only if $X$ is finite? If so, does anyone know of a non-Hausdorff UC topology on a countably infinite set?
- Is there known to be an equivalent condition on the cardinality of $X$ so that (2) $\implies$ (1)?
For your first question, the one-point compactification of the rationals is $KC$ (compact sets are closed) and hence $UC$, but it’s not Hausdorff.
Added: For the second question, suppose that $X$ is infinite. Fix $p\in X$, let $$\mathscr{B}=\big\{\{x\}:x\in X\setminus\{p\}\big\}\cup\big\{X\setminus F:F\subseteq X\setminus\{p\}\text{ is finite}\big\}\;,$$ and let $\tau$ be the topology generated by $\mathscr{B}$. Clearly $\langle X,\tau\rangle$ is a Hausdorff space, and it’s not discrete. Thus, (2) implies (1) only for finite spaces.