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?
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.