## Tag Archives: Homeomorphism

### A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.

Why?

Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. \$12.95 for the Dover edition (not sure when).

### An incorrect proof of the Jordan Curve Theorem – can you find what’s wrong with it?

Every closed curve in an infinite flat plane divides it into a bounded part and an unbounded part (inside and and outside if you’re not particular). This is so screamingly obvious, that for a long time no one thought it needed proof. Bolzano changed all that about 200 years ago, but a proof was not forthcoming until Jordan gave a proof (thought by most to be defective) in 1887.

The proof is long and subtle. The one I’ve read uses the Brouwer fixed point theorem, which itself uses the fact that fundamental group of a circle is infinite cyclic (and that’s just for openers). You begin to get the idea.

Imagine the 4 points (1,1),(1,-1),(-1,1) and (-1,1) the vertices of a square centered at ( 0, 0 ). Now connect the vertices by straight lines (no diagonals) and you have the border of the square (call it R).

We’re already several pages into the proof, when the author makes the statement that R “splits R^2 (the plane) into two components.”

It seemed to me that this is exactly what the Jordan Curve theorem is trying to prove. I wrote the author saying ‘why not claim victory and go home?.

I got the following back

“It is obvious that the ‘interior’ of a rectangle R is path connected. It is
only a bit less obvious – but still very easy – to show that the ‘exterior’
of R is also connected. The rest of the claim is to show that every path
alpha from a point alpha(O)=P inside the rectangle R to a point alpha(1)=Q
out of it must cross the boundary of R. The set of numbers S={alpha(i) :
alpha(k) is in interior(R) for every k≤i} is not empty (0 is there), and it
is bounded from above by 1. So j=supS exists. Then, since the exterior and
the interior of R are open, j must be on the boundary of R. So, the interior
and the exterior are separate components of R^2 \ R. So, there are two of
them.”

Well the rectangle is topologically equivalent (homeomorphic) to a circle.

So why isn’t this enough?  It isn’t ! !

Answer to follow in the next post. Here’s the link — go to the end of the post — https://luysii.wordpress.com/2015/11/10/are-you-sure-you-know-everything-your-protein-is-up-to/