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Seemingly the second fact is necessary for the proof of the first. However, it is not clear to me that metric structure is necessary for the proof of the first either (for example, why would uniform structure not be sufficient). This might, however, just be a result of the fact that I don't understand the concept of limit very well in spaces more general than metric spaces.

Question: What is the most general type of space for which a function is continuous if and only if it commutes with the limits of sequences?

Somewhere strictly between Hausdorff and metrizable spaces?

(Note: I'm very surprised that this question seems not to be a duplicate -- I could not find an equivalent question or the answer when Googling. If you know where to find the answer, please just close the question without downvoting and comment with the link.)

Background: Inspired by this question, in which an elegant proof using this property is given. Naturally I am curious to see the full generality to which this proof can be extended.

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Chill2Macht
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    Hausdorff spaces are T2. T1 spaces are too weak: http://math.stackexchange.com/questions/901154/unique-limits-in-t1-spaces -- this addresses a slightly different question, but consider that commuting the two operations may make the function ill-defined (having two distinct limits/values simultaneously is difficult for a function). This indicates that being Kolmogorov matters, so R1 ("preregular") isn't strong enough either. – Eric Towers May 08 '16 at 03:04
  • I'm not sure I get perfectly the question (for instance, do you means limits of nets?), but the wikipedia page about nets contains some relevant observations. –  May 08 '16 at 03:11
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    If $f:X\to Y$ where $X,Y$ are topological spaces and $X$ is First-countable then $f$ is continuous if and only if it is sequentially continuous. See http://math.stackexchange.com/questions/777865/proofs-about-continuity-and-convergence-in-topological-spaces?lq=1 – Dimitris May 08 '16 at 03:12
  • Oh oops I didn't realize that the answer would be different for nets and sequences. So basically: equivalence holds always for nets, but equivalence holds for sequences only for first-countable spaces? Does someone want to post this as an answer so I can upvote and accept it? Also thank you all very much for your help! – Chill2Macht May 08 '16 at 03:22
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    You can go a little beyond first-countable. See Pete Clark's Convergence, on sequences, nets and filters. He shows that (continuity = preservation of convergent sequences) for larger classes of spaces than first-countable — namely, Fréchet spaces (sequential closure of every subset is its closure), and what he calls sequential spaces. – BrianO May 08 '16 at 03:30
  • Thanks for the reference! This looks like really interesting reading -- it answers my question and discusses filters and ultrafilters (something I have been meaning to learn for a while now). – Chill2Macht May 08 '16 at 03:32

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