A Basic 5 Minute Crash Course into Basic Topology

Topology is a branch of Mathematics which generalizes many notions that we take for granted in geometry and calculus. Basic motivation here is that often times, when trying to solve a mathematical problem, it can be hard to see the forest for view of the trees (IE, when you consider too much information, a problem becomes overly complex and difficult to understand, ignoring some information that is irrelevate makes the problem easier).

Lets consider a motivating example: Suppose you are a vaccum cleaner salesman and have to cross 7 distinct bridges to reach 14 distinct houses- how many ways can this be done? What information do you need to solve this problem? This problem asks nothing about efficiency, or optimization. So really the only information you need is relative placement of bridges and houses- you need to know nothing about relative distances.

Topology can be viewed as Geometry without distance. We want to examine what properties of an object are preserved and maintained if we forget what distance is. For example, consider squares and circles:

A Square is defined to be a 4 sided polygon such that each side has equal length
A Circle is a simple closed curve such that each point on the curve is equidistanct from a common center.

Both of these definitions involve a notion of distance, now if we forget momentarily what distance is and look at a circle and a square side by side- how do we tell the difference between them?

To a Topologist(in general), there is no difference between a circle and a square, you can "deform" a circle into a square without breaking or cutting anything, so they are the same topologically (this is called being homeomorphic- the topological word for topologically identical).

Now, lets consider the difference between a circle and a figure 8. Here, a topologist can see a difference. You do not need any idea of distance to see that a circle has 1 hole and a figure 8 has 2 holes. You cannot morph a circle into a figure 8 (not without cutting the circle first and then glueing it back together in the form of a figure 8).

So to generalize, a topologist studies the types of properties which are preserved under these things called homeomorphisms. Basically this means, "can we deform this shape into some other shape without cutting it in some way"

This is a very basic crash course into the basic ideas behind General Topology, the overall field is very rich with many subfields addressing many problems.

Homework:

1) Visualize how to turn a coffee cup into a doughnut (classic topology problem), write step by step instructions on how to do this.

I must say first that this is my favorite group, and I truly enjoy it, although I'm far from being educated in math...

Excellent. I read one of Clifford Pickover's books about Klein bottles and Mobius strips, is that an ok start?

Only one way to fix that my friend, and time spent doing math is time spent well (of course this is my career so I am partial...)

I know, I'll get there...I'm planning on returning to school over the winter. At any rate, I'll be a frequent visitor here.

Klein bottles and Mobius stripes are excellent examples of some weird properties you encounter in Topology- namely non-orientable (and single sided) surfaces. Personally, I love these two objects!! Imagine yourself walking along a mobius strip, if you make one trip around the strip, your sense of right and left, once you complete the trip, will be reversed. If you want to try this, get a strip of paper and color one side of it black and the other white like this:

***************
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
*        *******
***************

So the coloring is done down the length of the strip. Now staple it together as a mobius strip- the black and white will not line up. Pretend black is your right and white is your left- they change!!

I am not familiar with the book you mention, but a fun one that involves a lot of scissors and paper, but does not require a lot of high power math to understand is Experiments in Topology by Stephen Barr:

http://www.amazon.com/Experiments-Topology-Stephen-Barr/dp/0486259331/ref=sr_1_1?ie=UTF8&s=books&qid=1257910192&sr=8-1

Topology by James Munkres is the gold standard intro textbook if you want to actually learn topology- I think you can get by with basic calculus and linear algebra and do ok with this book- but expect to work hard if you have had little exposure to formal higher math(I give this advice generally to anyone reading this).

A cheaper option is Introduction to Topology by Gamlin and Greene:

http://www.amazon.com/Introduction-Topology-Theodore-W-Gamelin/dp/0486406806/ref=sr_1_3?ie=UTF8&qid=1257910425&sr=1-3-fkmr0

This is actually the book I learned out of.

What are you studying?

Interesting! Although why one will see distance in a circle and not a figure eight makes it feel  like I am missing some intricate point here. One can take a circle and twist from the ends and make a figure eight. Thus using the same distance involved in a circle morphed into an eight. Ok, tell me what I am missing! Evidently I am no Topologist.

Math, Philosophy and English, (not necessarily in that order:)

The issue is not that you see distance in a circle but not a figure 8, the issue is you see no distance in either. You do not need a notion of distance to see that a hole is present. Imagine that you have a hole in the ground, if you were to put a spike in the hole, and then try to drag a piece of rope along the ground where the spike is, you would get stuck at the spike. This is how topologists detect holes on surfaces. They construct loops and curves and try to drag them around on the space. Anywhere a hole exists, the loop or curve cannot be dragged without breaking. One of the main tools behind this is whats called the Fundamental Group and is a major topic of a subfield of Topology called Algebraic Topology (This is the specific subfield of Topology that I intend to get my Ph.D. in).

Your point regarding twisting the circle is very valid (evidently you are more of a topologist than you think!). The intricate point you are missing is, in fact, one that I failed to mention (my bad). This thought experiment is being done in in 2 dimensions, so twisting from the ends is not possible.(IE, in a 2 dimentional world you have no height, so twisting has no meaning in this context).

What you are talking about in higher dimensions is kind of like another interesting field called Knot theory. The idea behind knot theory is similar to what I have outlined above.

A knot (in Knot theory) is basically a knot in the usual piece of string sense- but with the ends of the string glued together (so it is sort of like a tangled up circle). In knot theory, you try to figure out which knots can be "untied" into other knots.

See this video:

http://www.youtube.com/watch?v=AGLPbSMxSUM

Good Combo!

Awesome! Thanks Coldcoffee! I think you will make a great Doctor. You teach very well.  Didn't know if you would even answer me. Your not so cold after all.lol Very Interesting!

For the coffee cup-doughnut question, one would have to assume that the material was malleable, like clay. Then they would have to compress the clay to form a nearly flat ring, and finally work what was the handle and the bottom of the cup into that ring to make the final circle of the doughnut. I can understand the idea of topology very well with this example, I think.

(For a more modern approach, one could set a microwave oven to full power for a couple of hours or so, and probably end up with something close...just kidding. (kids don't try that at home!))

... may U just bring up that a coffee cup in a 2D world would be very different from one here.... I have now spent quite a lot of time drawing some crude pictures of 2D evolution and how coffee cups would end looking on paint, and now i cant upload them. Help?

stupid chess.com won't upload my pictures.

basicly i think a 2D coffee cup would have a handle like a "T" shape, not an "O"

I have a question if you are still tracking this, ColdCoffee. I read that some parts about Topology annoyed Lewis Carroll a lot and served as partial motivation for adventures in Alice and Wonderland. Do you know about this, particularly about methods with relationships between circles....