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I wanna ask why sequence is important in general topology. As far as i know, many theorem can be proved without using sequence. Does sequence make some proof easier than other way? or is there any other reason? If yes, can you give some example for references?

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Don't ask what sequences can do for general topology, ask what general topology can do for sequences. For general topology, analyzing convergence by sequences is not enough. The two ain approaches work with nets or filters instead. – Michael Greinecker Nov 18 '12 at 11:19
First you should know the topology is fully characterised by nets/filters. Well, although some might argue sequences are no longer important for general topological spaces, I do feel they are useful for they give a somewhat dynamical feeling about the space. Although I write down nets/ filters on exams, I think about sequences (nets are too large for my poor imagination). – Hui Yu Nov 18 '12 at 14:32
My topology classes used the Bourbaki approach. Nothing but filters and ultrafilters. I had no idea what was going on until I finally figured out how filters relate to sequences. – bubba Nov 19 '12 at 0:36

In formal rigorous terms, sequences are not important in general topology. As the comment said, you have to use filters or nets for rigorous general proofs. But sequences are useful because they are much easier to visualize than filters or nets, so they help your intuition. You might use sequences as examples, to figure out (roughly) how to prove something, but then you would use filters or nets for the final proof.

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