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Addendum: Here are some further simple considerations which unify some of the other examples given.

For a topological space $X$, consider the specialization relation: a point $x$ specializes to the point $y$ if $y$ lies in the closure of ${x}$. This implies that any sequence which converges to $x$ also converges to $y$. (If in the previous sentence we replace "sequence" by "net", we get a characterization of the specialization relation.) The specialization relation is always reflexive and transitive, so is a quasi-order.

Note that a topological space is T_1, or separated, iff the specalization relation is simply equality. Thus in a space which is not separated, there exist distinct points $x$ and $y$ such that every net which converges to $x$ also converges to $y$. If $X$ is first countable, we may replace "net" by "sequence".

A topological space $X$ satisfies the T_0 separation axiom, or is a Kolmogorov space, if for any distinct points $x,y \in X$, there is an open set containing exactly one of $x$ and $y$. A space is Kolmogorov iff the specalization relation is anti-symmetric, i.e., is a partial ordering. Thus in a non-Kolmogorov space, there exist distinct points $x$ and $y$ such that a net converges to $x$ iff it converges to $Y$. (If $X$ is first countable...)

An example of a first countable non-Kolmogorov space is a pseudo-metric space which is not a metric space (a pseudo-metric is like a metric except $\rho(x,y) = 0 \iff x = y$ is weakened to $\rho(x,x) = 0$). In particular, the topology defined by a semi-norm which is not a norm always gives such examples.

1

Here are two relevant facts:

1) In a Hausdorff space, a sequence converges to at most one point.

2) A first-countable space in which each sequence converges to at most one point is Hausdorff.

See e.g. pages 4 to 5 of

http://math.uga.edu/~pete/convergence.pdf

for the (easy) proofs of these facts, together with the definition of first-countable. See p. 6 for an example showing that 2) does not hold with the hypothesis of first-countability dropped.

It seems like a worthwhile exercise to use 2) to find spaces that have the property you want. For instance, the cofinite topology on a countably infinite set is first-countable and not Hausdorff, so there must be non-uniquely convergent sequences.