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When I was originally introduced to topology, I simply accepted most of its properties as generalizations of ℝⁿ. I didn’t give it any serious thought until about a month ago when I read an excellent thread on math overflow about it. Since then, its been one of the things I often find myself thinking about when I’m trying to fall asleep. Given the amount of thought I’ve put into it, and the fact that I feel I should be answer questions like this about topology, given that it’s one of the areas of math I spend a lot of time on, I thought I’d write up my thoughts. They lent themselves well to being written in the form of an introduction to topology, so that’s what I did.
(After finishing this essay I decided to reread the MO thread. The first comment — not answer, a comment — mentions the Kuratowski closure axioms and closure axioms sounded like one might call what I came up with. Sure enough, they’re the exact same, down to the ordering. Are all attempts to make mathematical contribution’s this frustrating? I’m posting this because of the amount of work I put in, but there’s nothing new here.)
Consider 1 with respect to [0,1). It isn’t part of the set, but in a sort of intuitive sense it almost is. And knowing which points are `almost in’ a set gives us lots of information, for example notions of boundaries and connectedness. Topology is based on us formalizing this notion of `almost in’ and once we formalize it, we can consider non-standard notions of being `almost in’ or apply these ideas to spaces that we don’t typically associate them with. (more…)