I remember looking at the course schedule when I was near the end of my first semester, senior year. I noticed that a “topology” course was being offered; I also remember reading a bit about “topology” in the Time-Life book on mathematics. I remembered a donut shaped object (called torus), Klein Bottles, Mobius bands and the like. I wondered if the course would be a sequence of parlor tricks.
Then when I got to the course; well….it wasn’t what I expected.
Ok, what is going on? I heard terms like “open”, “closed”, “basis”, “Hausdorf”, “Regular”, “Normal”, “second countable”, “first countable” etc. It reminded me of “analysis on steroids”. I began to wonder if the Time-Life book was just making it all up.
But it wasn’t; toward the back of our text (Munkres) there was even a drawing of a double torus.
So, what in the world is going on?
Well, I’ll just write informally; if you can catch my omissions you already know this stuff. 🙂
Probably the most basic question topology asks is: “when are two spaces “the same””. So, what do you mean by “the same”?
In topology the answer is almost always: given spaces is there a continuous bijection (one to one and onto function) which as a continuous inverse ? If there is such a function, the spaces are said to be “topologically equivalent” or “homeomorphic”. You might notice the term “continuous” and wonder what it means in an abstract context like this.
The usual calculus “epsilon-delta” definition works if are real n-spaces with the usual open intervals/disks/balls, etc.
So here is an elementary example: the unit interval is homeomorphic to any other closed interval . Here is the proof: the map (called a “homeomorphism”) is given by . Notice that is a bijection and has a continuous inverse. On the other hand, is NOT homeomorphic to . The standard way to see the latter is to use the tools developed in either analysis courses or a beginning topology course that I am describing; the quick answer is that the closed interval is “compact” whereas the half-open interval is not. Or, another way: has all of its limit points; is missing a limit point.
You go on to talk about what an open set is. The calculus notion is that an open set is one that is built up as the collection of open intervals (in the real line) or open disks, open balls, etc. In topology, you take a collection of sets and if this collection meets the following properties: the whole space and the empty sets belong to and if an arbitrary union of subsets of belong to and any FINITE intersection of elements of belong to , then is said to be a topology for the space.
Clearly, the standard “open” intervals form a topology for the real line; we do calculus with these. But one can form a topology generated by, say, half open intervals (these things are bizarre) or other stranger collection of sets.
Yes, the complement of an open set is a “closed set”, and yes, in some topological spaces, there are sets that are both open and closed at the same time. (grrr…) Example: given a set , declare EVERY point in to be an open set.
It is the study of these things that often constitute the first part of a first topology course. And yes, you DO need to know this stuff.
So what about the geometric stuff?
Well, let’s start small. You know that there is a bijection between and the unit circle (the bijection is . You also know that and the unit circle are both compact sets (think: “closed and bounded” if you are new). But they are NOT homeomorphic sets. Learning why starts you in the more geometric direction. One easy way: if you look at the intervals, removal of any point except the end points separates the intervals into two pieces, where no one point separates the unit circle into two pieces. That is a more “global” property.
So in this more global view, you’ll learn not only geometric type arguments but also how do use algebra (yes, at advanced levels, math is not as compartmentalized as it appears to many undergraduates). That is part of algebraic topology.
And, of course, to simplify the types of objects studied, one might want to but a differential structure on a space (assign the notion of a derivative and “tangents”) by attaching something called the “tangent bundle” to a space. That is the subject of differential topology. Here the homeomorphisms are often required to be infinitely differentiable as well.
So, yes, there IS a connection between the “rubber sheet” geometry stuff that you read about in the popular media and the abstract sounding stuff that you sometimes get at the start of an undergraduate topology class. It just takes a bit of time and effort to get there.
Now the type of topology that really never gets to the geometric stuff is called “point set” topology; there is something called “general topology” too. (don’t ask). 🙂