A Taste of Topology
Jul. 15th, 2009 04:45 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowAs I've mentioned before, this summer I'm teaching a course in topology. One of the pleasures of teaching a new course, especially an advanced one, is re-encountering material you (in theory) learned many years ago, but seeing it with a more mature and experienced eye. Last week, we covered "Urysohn's Lemma", and I found to my delight that its proof (long since submerged) is, well, really neat.
I'm going to spend a post or three laying out enough background that I can give the basic structure of the proof, without going into all the grungy details. Look under the cut... if you dare....
First off: if images of Möbius strips and deformed doughnuts are dancing in your head, set them aside. They are part of topology, but unfortunately the stretch between what I'm going to talk about and that material is long and daunting, and I'm not going there. Sorry.
The basic inspiration for topology is, roughly speaking, the notion of nearness. (The dictum against "tearing" things arises from the fact that a tear pulls apart points which are very close together, no matter how stringently you define "very close" - within 0.1? Within 0.01? Within 0.0000000000000001?) There was much discussion, a century or so ago, on how to capture this notion, and what ultimately emerged is the following.
A topological space X is simply a set, some of whose subsets are designated as "open". Roughly speaking, a subset S is open if, for every x in S, every point which is "close enough" to x is also in S. So, for example, in the real numbers, an interval like (0,1) is open. (0.9 is in the interval; every number which is less than 0.1 away from 0.9 is also in the interval. 0.01 is in the interval, and every number within, say, 0.005 of 0.01 is also in the interval.) An interval like [0,1] is not open, because you can get as close as you like to 1 without actually entering the interval - just come in from above.
The choice of which subsets are open isn't completely arbitrary; they have to satisfy certain conditions. First, the empty set and X have to be open. Second, the intersection of any two open sets must be open. Third, the union of any collection, finite or infinite, of open sets is also open. That's all.
Once you've decided what the open sets are, you can also identify the closed sets. "Closed" doesn't mean "not open"; a set F is closed if its complement is open. That is, F is closed if, whenever x is not in F, neither are any of the points that are close to x. (The interval [0,1] is closed; 1.01 is not in the interval, for instance, and neither is any number within, say, 0.001 of 1.01.) Note that "closed" doesn't mean "not open"; it's possible for a set to be both, either but not the other, or neither. The empty set, for instance, is always both open and closed. In the real numbers, an interval like [0,1) is neither open nor closed.
One final definition, and I'll close this first segment. If S is any subset of X, the closure of S is the smallest closed set that contains S. For example, in the real numbers the closure of (0,1) is [0,1]; the closure of the set of rational numbers is the entire set of real numbers. (If you pick any number r, rational or irrational number, and set any degree of closeness, there's a rational number that's that close to r. Remember Cauchy and Dedekind?)
Next up: some examples, familiar, exotic, and pathological, and how we avoid the pathological cases.
I'm going to spend a post or three laying out enough background that I can give the basic structure of the proof, without going into all the grungy details. Look under the cut... if you dare....
First off: if images of Möbius strips and deformed doughnuts are dancing in your head, set them aside. They are part of topology, but unfortunately the stretch between what I'm going to talk about and that material is long and daunting, and I'm not going there. Sorry.
The basic inspiration for topology is, roughly speaking, the notion of nearness. (The dictum against "tearing" things arises from the fact that a tear pulls apart points which are very close together, no matter how stringently you define "very close" - within 0.1? Within 0.01? Within 0.0000000000000001?) There was much discussion, a century or so ago, on how to capture this notion, and what ultimately emerged is the following.
A topological space X is simply a set, some of whose subsets are designated as "open". Roughly speaking, a subset S is open if, for every x in S, every point which is "close enough" to x is also in S. So, for example, in the real numbers, an interval like (0,1) is open. (0.9 is in the interval; every number which is less than 0.1 away from 0.9 is also in the interval. 0.01 is in the interval, and every number within, say, 0.005 of 0.01 is also in the interval.) An interval like [0,1] is not open, because you can get as close as you like to 1 without actually entering the interval - just come in from above.
The choice of which subsets are open isn't completely arbitrary; they have to satisfy certain conditions. First, the empty set and X have to be open. Second, the intersection of any two open sets must be open. Third, the union of any collection, finite or infinite, of open sets is also open. That's all.
Once you've decided what the open sets are, you can also identify the closed sets. "Closed" doesn't mean "not open"; a set F is closed if its complement is open. That is, F is closed if, whenever x is not in F, neither are any of the points that are close to x. (The interval [0,1] is closed; 1.01 is not in the interval, for instance, and neither is any number within, say, 0.001 of 1.01.) Note that "closed" doesn't mean "not open"; it's possible for a set to be both, either but not the other, or neither. The empty set, for instance, is always both open and closed. In the real numbers, an interval like [0,1) is neither open nor closed.
One final definition, and I'll close this first segment. If S is any subset of X, the closure of S is the smallest closed set that contains S. For example, in the real numbers the closure of (0,1) is [0,1]; the closure of the set of rational numbers is the entire set of real numbers. (If you pick any number r, rational or irrational number, and set any degree of closeness, there's a rational number that's that close to r. Remember Cauchy and Dedekind?)
Next up: some examples, familiar, exotic, and pathological, and how we avoid the pathological cases.
