General > RANT
Continuance of the INfinity Proposal
abyaly:
So far, there has been much more mention of balls in real analysis than topology. In general, I don't think it makes sense to apply the word geometry to something without a notion of distance.
Numsgil:
It doesn't talk about distance directly, but abstracts the concept into a general function relating two elements in a set called a metric. See metric @ wiki.
abyaly:
A topological space may not be (and very often isnt) a metric space. That is, there is no metric that can be used to generate the topology. Metrics are another thing that real analysis has talked a lot about and topology has mentioned not at all.
Numsgil:
A metric space is a topological space (though the reverse isn't necessarily true). I learned about metrics and metric spaces in my topology class. Though to be fair I never finished the real analysis class, and didn't do particularly well in my topology class either. But it sounds like we disagree more just on how the definition is used than anything substantial. In this post when I talked about topology I meant specifically the branch of topology that deals with euclidean spaces and related non flat metric spaces that can be used to describe possible space times.
BTW, if you know enough to argue with me... were you a Math major? I can't imagine any other reason anyone would ever learn or care about real analysis or topology.
abyaly:
I'm currently in my first year of grad school in math. When I mentioned what real analysis and topology talked about, I was using the terms as names of classes rather than names of mathematical fields. I don't disagree with how you used metric, but I don't think calling topology a type of geometry fits because metric spaces are only a part of it. Although I'm fine with calling abstract geometry a type of topology :-P
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