College Math Teaching

February 18, 2015

Another way to make a math professor happy…

Filed under: editorial — Tags: — collegemathteaching @ 9:29 pm

One of my topology students came in and pointed out (very respectfully) an error on the online course notes. I had written that f(x) = \frac{1}{x} is a homeomorphism between (0,1) and (0, \infty) and she pointed out that the second interval should be (1, \infty) .

I am doing a happy jig…. 🙂

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February 16, 2015

Topologist’s Sine Curve: connected but not path connected.

Filed under: student learning, topology — Tags: , , — collegemathteaching @ 1:01 am

I wrote the following notes for elementary topology class here. Note: they know about metric spaces but not about general topological spaces; we just covered “connected sets”.

topsincurve

I’d like to make one concession to practicality (relatively speaking). When it comes to showing that a space is path connected, we need only show that, given any points x,y \in X there exists f: [a,b] \rightarrow X where f is continuous and f(a) = x, f(b) = y . Here is why: s: [0,1] \rightarrow [a,b] by s(t) = a + (b-a)t maps [0,1] to [a,b] homeomorphically provided b \neq a and so f \circ s provides the required continuous function from [0,1] into X .

Now let us discuss the topologist’s sine curve. As usual, we use the standard metric in R^2 and the subspace topology.

Let S = \{(t, sin(\frac{1}{t}) | t \in (0, \frac{1}{\pi} \} . See the above figure for an illustration. S is path connected as, given any two points (x_1, sin(\frac{1}{x_1}), (x_2, sin(\frac{1}{x_2}) in S , then f(x) = (x, sin(\frac{1}{x}) is the required continuous function [x_1, x_2] \rightarrow S . Therefore S is connected as well.

Note that (0,0) is a limit point for S though (0,0) \notin S .

Exercise: what other limit points does S that are disjoint from S ?

Now let T = S \cup \{ (0,0) \} , that is, we add in the point at the origin.

Fact: T is connected. This follows from a result that we proved earlier but here is how a “from scratch” proof goes: if there were open sets U, V in R^2 that separated T in the subspace topology, every point of S would have to lie in one of these, say U because S is connected. So the only point of T that could lie in V would be (0,0) which is impossible, as every open set containing (0,0) hits a point (actually, uncountably many) of S .

Now we show that T is NOT path connected. To do this, we show that there can be no continuous function f: [0, \frac{1}{\pi}] \rightarrow T where f(0) = (0,0), f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0 ) .

One should be patient with this proof. It will go in the following stages: first we show that any such function f must include EVERY point of S in its image and then we show that such a function cannot be extended to be continuous at (0,0) .

First step: for every (z, sin(\frac{1}{z})), there exists x \in (0,\frac{1}{\pi} ] where f(x) = (z, sin(\frac{1}{z}) ) Suppose one point was missed; let z_0 denote the least upper bound of all x coordinates of points that are not in the image of f . By design z_0 \neq \frac{1}{\pi} (why: continuity and the fact that f(\frac{1}{\pi}) = (\frac{1}{\pi}, 0) ) So (z_0, sin(\frac{1}{z_0}) cuts the image of TS into two disjoint open sets U_1, V_1 (in the subspace topology): that part with x-coordinate less than and that part with x-coordinate greater than x = z_0 . So f^{-1}(U_1) and f^{-1}(V_1) form separating open sets for [0,\frac{1}{\pi}] which is impossible.

Note: if you don’t see the second open set in the picture, note that for all (w, sin(\frac{1}{w})), w > z_0 one can find and open disk that misses the part of the graph that occurs “before” the x coordinate z_0 . The union of these open disks (an uncountable union) plus an open disk around (0,0) forms V_1 ; remember that an arbitrary union of open sets is open.

Second step: Now we know that every point of S is hit by f . Now we can find the sequence a_n \in f^{-1}(\frac{1}{n \pi}, 0)) and note that a_n \rightarrow 0 in [0, \frac{1}{\pi}] . But we can also find b_n \in f^{-1}(\frac{2}{1 + 4n \pi}, 1) where b_n \rightarrow 0 in [0, \frac{1}{\pi}] . So we have two sequences in the domain converging to the same number but going to different values after applying f. That is impossible if f is continuous.

This gives us another classification result: T and [0,1] are not topologically equivalent as T is not path connected.

February 14, 2015

No, I don’t “learn more from my students than they do from me”: BUT…..

I admit that I chuckled when a famous stand up comic said: “”New Rule: Any teacher that says, ‘I learn as much from my students as they learn from me,’ is a sh***y teacher and must be fired.””

Yes, I assure you, when it comes to subject matter, my students had bloody well learn more from me than I do from them. 🙂

BUT: when it comes to class preparation, I find myself learning a surprising amount of material, even when I’ve taught the class before.
For example, teaching third semester calculus (multi-variable) lead me to thinking about some issues and to my rediscovering some theorems presented a long time ago and often not used in calculus/advanced calculus books. THAT lead to a couple of published papers.

And, given that my teaching specialty has morphed into applied mathematics, teaching numerical analysis has lead me to learn some interesting stuff for the first time; it has filled some of the “set of measure infinity” gaps in my mathematical education.

So, ok, this semester I am teaching elementary topology. Surely, I’d learn nothing new though I’d enjoy myself. It turns out: that isn’t the case. Very often I find myself starting to give a proof of something and find myself making (correct) assumptions that, well, I last proved 30 years ago. Then I ask myself: “now, just why is this true again?”

One of the fun projects is showing that the topologist’s sine curve is connected but not path connected (if one adds the vertical segment at x = 0). It turns out that this proof is pretty easy, BUT…I found myself asking “why is this detail true?” a ton of times. I drove myself crazy.

Note: later today I’ll give my favorite proof; it uses the sequential definition of continuity and the subspace topology; both of these concepts are new to my students and so it is helpful to find reasons to use them, even if these aren’t the most mathematically elegant ways to do the proof.

This is why I proved the Intermediate Value Theorem using the “least upper bound” concept instead of using connectivity. The more they use a new concept, the better they understand it.

February 11, 2015

Teaching advanced stuff to undergraduates: taking stuff for granted….

Filed under: advanced mathematics, pedagogy, point set topology — Tags: , — collegemathteaching @ 5:34 pm

I am teaching undergraduate topology this semester; you can see some stuff that I’ve posted for undergraduates here.

I took the “start with metric spaces and topology of R, R^2, R^3.. approach and am going slower than I’d like. But it takes some time to absorb the stuff.

So, we are finally gotten up to homeomorphisms (still in basic metric spaces) and so I figured that we were finally ready to show something like: [0,1] is not homeomorphic to S^1 (the unit circle). Fine: suppose a homeomorphism exists and decompose [0,1] =U \cup V\ \{x \} where x \notin U, x\notin V, U \cap V = \emptyset

Not a problem so far…so now pull back the disjoint open sets U, V to the unit circle minus one point…and….then…I ….realized….that…I have not proven that the interval is a connected set; in fact I haven’t even defined “connected set”. &^%$#. Now, that isn’t that hard to do, but it does take time and one has to do some setting up.

And that is one problem with teaching this course: you have to do so much elementary stuff to even begin to prove something elementary.

February 2, 2015

The challenge of teaching undergraduate topology

Filed under: pedagogy, topology — Tags: — collegemathteaching @ 10:44 pm

I mentioned that I’d be teaching undergraduate topology for the first time ever. Yes, I’ve taught “proof required” courses before: on two separate occasions between 1991 and now. But in each case, I taught a senior level abstract algebra course in which every student came in having taken a course that required proof.

Technically speaking, this course doesn’t require a previous “proof” course.

Then there is how long I’ve been doing this; my first published paper appeared in 1991 (I did the work in 1989). That was well before any student in my class was born!!!

So, this process is going to be a bit like a native language speaker trying to teach someone else a new language.

This means: I’ll have to slow down; this book I’ve chosen won’t be too easy after all. 🙂 I am going to have to back way off of the pace. After all, I am teaching both the material AND proof writing.

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