Topology

Date: 03/19/2001 at 09:21:32
From: shumi
Subject: What is Topology?

Hello, Dr.Math.

I'm a college student in Taiwan, and I major in math. I haven't 
studied topology, but I want to know something agout it. Can you tell 
me what topology is?

Thanks!  :)

Date: 04/02/2001 at 10:09:25
From: Doctor Nbrooke
Subject: Re: What is Topology?

Good morning Shumi, and thanks for writing to Dr. Math.

Topology is the study of properties of objects that are preserved 
though twisting, squishing, and stretching, but not tearing. 

In topology, a circle is equivalent to an ellipse, since you can 
squish a circle into an ellipse and stretch an ellipse into a circle. 
They both have the same topological properties. You can change the 
geometry (basically the distance between points in an object) of an 
object by stretching it, but not its topology.  

When we study topology we learn which properties of an object are 
preserved through these operations, like its orientability and its 
genus (basically the number of "holes" in it). A doughnut and a coffee 
cup are topologically equivalent since they are both solid objects 
with only one hole; it is possible to deform a doughnut into a coffee 
cup without tearing, and vice versa.  

Once we know what types of properties we're looking for, we study the 
different types of objects, called manifolds, and how they relate to 
each other. We learn about tori, projective planes, Moebius strips, 
and Klein bottles - and that's only in two dimensions! You know that 
if you remove a point from a circle you get a line segment; well, if 
you remove a point from a Klein bottle you get two Moebius strips.

This is a very cursory introduction to the study of topology, but I 
hope it helps you to know what to expect soon. Please feel free to 
write back.

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/