Every so often I like to go back and read through some basic algebra or basic point-set topology and try to think of new ways to look at the easier, more fundamental problems.  I’ve been going through Bredon’s Algebraic Topology to prepare for a topology topics class next semester and there are two wonderful things about this book:

  1. It is concise*.  When you read Bredon and Hatcher back-to-back you’ll understand what I mean here: while I do love Hatcher’s long motivating paragraphs (and, in fact, I attempted to mimic his method of explaining things in depth when I first began this blog), I tend to get lost in his prose at times.  (On the other hand, Bredon’s book had significantly less motivation.  Significantly less.)
  2. The questions are few, but proud.  No one can deny that Hatcher’s book has a ton of enlightening and challenging questions.  On the other hand, Bredon’s questions are generally more focused toward the section immediately preceding it (helping the beginning student) and are generally quite interesting (which is not to say the Hatcher ones are not!).  The nice part about Bredon’s book is that you could easily assign students a section and ALL of the associated questions and they would probably not complain too much since there’s usually only three or four associated questions.  Not so with Hatcher.

(*Perhaps not as concise as May’s book, though!)

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The Sierpinski Space.

March 22, 2011

What’s the weakest separation we can have in a topological space?

Well, “no separation” is pretty weak.  But this creates the trivial topology and that’s a bit boring.  So let’s say this:


Definition: A topological space X is T_{0} or Kolmogorov if for every two points x,y\in X we have that there exists a neighborhood U such that either x\in U and y \notin U or y\in U and x\notin U.


In other words, a space is T_{0} if for every pair of points there is at least one open set which contains one and doesn’t contain the other.  This is a pretty weak separation condition.  Certainly, every Hausdorff space is T_{0}, but there are ones which are even weaker which satisfy this condition.  Let’s try to construct a really easy one.

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I do this post anticipating the question, "Yeah, but why are these formulas the same?" from some of my statistics students.  This is all done in the discrete random variable sense. 

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I cannot find a suitable picture of a linked double torus, which makes me think it’s called something else.   Luckily, because it’s on the cover of Armstrong’s book (and I don’t think Springer would mind some free advertising) we can reproduce it here:


Now, how can we take this "linked" double torus and make it an "unlinked" double torus without cutting anything up?  This one kept me up last night, but the solution turns out to be quite easy once you see it. 

If you don’t want the full solution, a hint might be: start thinking of the double torus as two handles on a sphere.  What can you do with these handles?


The picture solution after the jump.

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Small Anecdote.

When I was going through my first proofs class it was my first year in college — needless to say, my mind wasn’t completely in the game.  Because it was a Moore method class (the students would do the proofs on the board, and the students would critique it.  the professors were there to help us stay on task and to provide additional criticism.) a number of students would simply copy proofs from Spivak and present them as their own — and this caused a large number of students (including myself, I’ll shamefully admit) to be in possession of a large number of proofs that they have "written" but did not understand. 

For me, a notable exception was the proof of the Bolzano-Weierstrauss property.  This was not only my favorite theorem at the time (and still is one of my favorites!) but it was one where I did the proof entirely by myself (on a bus, an hour before class).

Now-a-days when I am lying in bed and I’m too sleepy or too cold to move from under my covers, I like to reach around my bed and pick up a math book and read one of the sections.  Recently, after cleaning my room, I had moved my Hatcher which usually sits faithfully at my bedside to the other side of the room; getting up was unthinkable, so, instead, I picked up Armstrong’s Basic Undergraduate Topology.  In it, I found a slick proof of the Bolzano-Weierstrauss property that I haven’t seen before.  So, I thought I’d write it up.


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