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Following this post Topological properties preserved by continuous maps, we see that continous functions preserve only some topological properties. My question is, from these some, is there any collection of them such that it automatically implies a give map is continous, provided that this map preserves those properties?

Clemens Bartholdy
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  • Do you have any candidates or efforts to contribute? – Ted Shifrin May 22 '24 at 04:28
  • As an example, assuming the domain and codomain of $f$ are both metrizable (or really, the domain of $f$ being sequential and the codomain being Hausdorff would be enough), then preserving the collection of convergent sequences would be enough to imply $f$ is continuous. – David Gao May 22 '24 at 04:34
  • I genuienly believe that this question is above my current abilities and time constraints to pursue further on my own. @TedShifrin – Clemens Bartholdy May 22 '24 at 04:39
  • A remark: the example I mentioned in comments under Jonas’s answer, i.e., the identity map from the space of two points with trivial topology to the Sierpiński space shows: preserving the collection of convergent sequences, preserving the collection of convergent nets, preserving connected subspaces, being open, being closed, and preserving compact subspaces are all insufficient to ensure continuity. – David Gao May 22 '24 at 20:01

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A map $f:X\rightarrow Y$ between topological spaces $X$ and $Y$ is continuous if and only if it preserves all those limits of nets, which exist in $X$. See Wikipedia. It is the topologically correct notion of convergence (sequential limits are often not enough), and is the proper thing to write on the last bullet of the cited question.

Jonas Linssen
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  • It should be noted that $f$ has to actually preserves the limits, not just what nets are convergent. For example, in both the trivial topology on two points and in the Sierpiński space topology, all nets are convergent, but the identity map from the space of two points with trivial topology to the Sierpiński space is not continuous. – David Gao May 22 '24 at 19:31
  • Sure. Do you think I can improve upon my answer (is it not clearly phrased?) or did you just want to stress that point again? – Jonas Linssen May 22 '24 at 19:33
  • In the question the OP linked to, the last bullet point seems to be only about sending convergent sequences to convergent sequences, not about preserving the limits. As you mentioned the last bullet point in your answer, I feel like it’s better to specifically stress this point. (Perhaps an additional remark that can be added is that if the codomain is $T_1$, then preserving just what nets are convergent, without a priori knowing the limits are also preserved, is actually enough to conclude the map is continuous.) – David Gao May 22 '24 at 19:41
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    Ahh thank you, I actually misread "preserving convergent sequences" as meaning "preserving limits of convergent sequences". I didn't even think it was an option to just ask for the property of being convergent to be preserved. – Jonas Linssen May 22 '24 at 19:55