If $X$ can be described by sequences, then it is metrizable

Question

Does anybody know about a result like "If $X$ can be described by sequences then it is metrizable"? By "described by sequences" I mean (I think I do, not sure yet) that the topology of $X$ is given by a sequence convergence notion.

See my last comment in the accepted answer to this question. There, @JohnGriffin comments about such a result. In his words,

"If $X$ can be described by sequences (in particular, it is metrizable), then [...]"

But, if the sequence convergence notion does not have the limit uniqueness property, then $X$ cannot be metrizable, since metrizability implies Hausdorff.

So, what would be a more precise statement? Which hypothesis, others than limit uniqueness do we add to sequentially described to get metrizability? It would be very helpful to have a result in that direction.

Answer 1

The statement under any reasonable interpretation is not true. They are called sequential spaces in the literature, and there exists Hausdorff sequential spaces that are not metrisable.

One would expect this is the case because first countable spaces are sequential, which have a much weak condition of having a countable neighborhood base (in fact, a space is sequential if and only if it is the quotient of a first countable space).

For an explicit counterexample, consider the real line with the lower limit topology. This is first countable and Hausdorff (and even paracompact), but it is not metrisable being separable but not second countable.

See also this post which discusses sequential spaces in some more detail (in particular talks about sequential continuity).

Answer 2

There are plenty of metrisation theorems that give necessary and sufficient conditions for a space $X$ to be metrisable. But none that I know of use sequences. IMO, A first countable Hausdorff space is the best-behaved general topology approximation of the sequence properties we know of metric spaces:

• A set is is sequentially closed iff is is closed.
• A function defined on it is sequentially continuous iff it is continuous,
• Sequences have unique limits,
• Being sequentially compact is equivalent to being countably compact and limit point compact.
• This class of spaces is closed under countable products and subspaces.
• If $X$ is a topological group (so it already has a lot of structure) then $X$ is metrisable iff it is first countable Hausdorff.

But such spaces can be far from metrisable, e.g. the Sorgenfrey line ($\mathbb{R}$ in the lower limit topology in Munkres) and the Double Arrow ($[0,1] \times \{0,1\}$ in the order topology from the lexicographic order) are first countable Hausdorff spaces with lots of extra nice properties (like Lindelöfness/compactness resp. and complete normality etc.) that are not metrisable. But from the viewpoint of just sequential convergence, they are indistinguishable from metric ones, I believe. If you can find a purely sequential property $P$ that holds in metric spaces but not in first countable Hausdorff spaces, leave a comment. I'd be surprised.