Saturday, September 8, 2012
只今！
So I just got back from Japan a little bit ago. Sorry for the lack of updates on that, but I had little to no internet access over the last couple months, but now I have good internets again and also a bunch of pictures of the trip, although they are mostly pictures of me making faces in front of stuff, so I will get to posting that tomorrow or later. As for now I must get my body back on EST.
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5 comments:
Hey there! way to go on Japanification - I'm jealous. We're all looking forward to plenty of photos of you doing the peace-sign in front of all sorts of landmarks and beautiful views. I thank you, too - your absence inspired me, uhh, not to blog all summer. Excellent.
Warping back to the 400-level topology and "geometry" classes (of which I only took the second, oops) of senior year at The Wez, I have a class-quirk question: did we lazily define open as not closed, or was it vice versa? There are so many ways to denote the two that I don't remember which cop-out definition we used.
(
I get the feeling we chose "closed" = "contains boundary" and "open" = "isn't closed," but I also get the feeling that such a super-simple rendition of the definition is skipping some technicalities. Contrary to what I appear to have established, though, I promise I do remember quite a bit of that material...
)
Ha, you might enjoy that I taught some topo when I was in the hospital last winter... One of the more interesting almost-a-doctor residents was really friendly to me for whatever reason, and he usually hung out in my room for a little while extra to schmooze on his rounds. He figured out that I was a math and compsci person pretty early and was interested, so he kept asking me to "teach" him some "fancy math," as he called it; I, as a thoroughly medicated algebra addict, was ready to break out mod-arith/rings/groups/isomorphisms/etc, but I knew I'd get too excited and in-depth and not share any of the "parlor trick" math bits most people want to know. So, I explained some bare-bones definitions and properties of closure, boundaries, and mega-abstract geometry in R^3. Then, as was the secret purpose of the "lesson," I wrapped up with the coffee-cup-and-donut mistaken-topologist zinger - and he got it! Score. Such was not expected.
You might also enjoy: "canonical." QED
***NOTE TO HR DEPARTMENTS, PROFESSORS, AND OTHER MATH HIGHER-UPS WHO HAVE READ THIS FAR: Even on post-op medications, I successfully taught senior-level math concepts to a medical doctor. Please hire me, accept me into your M.A. or Ph.D. programs (I was an RA at undergrad, and I'm willing to do it again!), and/or hire me as a professor.
tl;dr math math math, blah blah blah.
I want to say we probably defined closed as the complement of an open, but maybe we started with closed as containing its boundary? I don't think so just because it's hard to define what a boundary is without reference to open sets. I think we probably used the typical three axioms of open sets for defining a topology, since I don't remember using Kuratowski's notion of a closure operator. This requires a whole post, I think! But first, Japan pics.
good point concerning actually defining the boundary; I'll put in some thought time on that.
this is kind of frivolous, so don't waste much (or any) of your quite valuable time on the matter: what were those three axioms again? j'ai oublié (quite pathetically.)
thanks for the attention. keep up the awesome.
Hardly a waste of time, the axioms are, for a space X,
1) X and the empty set are open
2) If two sets are open, so is their intersection
3) If every set in an arbitrary family of sets is open, so is their union
Phrased another way, 2) and 3) say that topologies are closed under finite intersections and arbitrary unions.
Usually the boundary of A is defined as the set of all points such that every neighborhood intersects A and X/A non-trivially, i.e., neither of those intersections is empty. You can probably dance around that starting with closed sets and defining the closure of a set to be the smallest closed set containing it, then intersect the closures of a set and its complement to get the boundary or some such thing. I'd have to do some basic set theoretic computation to check it, but meh, something like that.
massively useful. thanks a ton. I wonder if I still have that "textbook" from Topics/Geometry; might be worth a flip-through. it'd be more useful if I had the text from the first-sem Jeter topo class. continuing the parallel structure progression, it'd be most useful if I had actually taken both semesters. oh well, nothing a little time on wiki-p and itunes u can't fix. encore, merci for the brush-up.
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