It is often stated that a metric topology is 'fully described by its convergent sequences'. How can this be expressed in a more formal way?
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The formal statement would be
Theorem. Two metrics on a set determine the same topology if and only if they have the same convergent sequences.
Ethan Bolker
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Does this also hold for any two Hausdorff spaces? – Danny Duberstein Nov 27 '20 at 17:12
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Very likely. I haven't thought about this kind of thing since grad school many years ago. Why not try to write a proof? I suspect it's straightforward. – Ethan Bolker Nov 27 '20 at 17:19
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Turns out not to be true: https://math.stackexchange.com/a/76700/435462 – Danny Duberstein Nov 27 '20 at 17:26
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In $\ell^1$ a sequence converges in norm if and only if it converges in the weak topology but the weak topology is strictly weaker than the norm topology. So, in general topological spaces, sequences fail to describe completely the space. If you use nets instead of sequences then it's true. – Evangelopoulos Foivos Nov 27 '20 at 17:31
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@DannyDuberstein: The spaces whose topologies are completely determined by their convergent sequences are the sequential spaces. – Brian M. Scott Nov 27 '20 at 18:35
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@EvangelopoulosF. Thanks for reminding me that you need to think about nets, not just sequences. – Ethan Bolker Nov 27 '20 at 21:43