By Mike Stannett
[See also: Introduction to General Topology]
What is general topology?
[See also: Introduction to General Topology]
General topology is that branch of pure mathematics that concerns itself with the nature of continuity.
What does it mean for a function f: X®Y to be continuous? The usual
answer is that when two points x and y are 'close together' in X, the values of
f(x) and f(y) should also be close together in Y. But this answer is rather
unsatisfactory because it brings in the idea that we need to measure distances - and that means introducing
an otherwise irrelevant space like R, the real line - we need to
express what these distances are, and all standard techniques for doing so require the use of numbers.
But continuity is such a fundamental property that it ought to be possible to define it simply in terms of X and Y themselves. Topologists discovered that the crucial concept was that of the open set. Given an arbitrary set X, we assemble a collection Á of subsets (this can be done almost at random). The collection has to contain both X itself, and the empty set. Then we add as many sets to Á as we need to to make sure that the following conditions are satisfied:
A collection of sets satisfying these conditions is called a topology on X, and the members of the collection are called the open sets of the topology. If U is an open set in X, we call its complement X\U a closed set. It's entirely possible for a subset to be both open and closed. We usually call such sets clopen. If A is a subset of X, the intersection of all of the closed sets which contain A is itself closed, so it's the smallest closed set containing A. We call it the closure of A.
Because we have so much freedom in choosing the open sets, it's possible for one and the same set X to be given lots of different topologies, and we should really write something like (X,Á) to remind ourselves that the topology and the underlying set are different things. In general, however, we usually just talk about X and assume that the topology can be taken for granted. In particular, there are some spaces which are used so frequently in mathematics that we always assume they carry the same standard topology. Examples include the real line R and some of its subsets, like the set Q of rational numbers (fractions).
To a topologist, a function f: X ® Y is continuous provided f -1(V) is open in X whenever V is open in Y. It turns out that this definition is equivalent to the older distance based version for all the standard spaces, but can be applied much more generally. There are spaces for which the concept of distance is utterly meaningless, and yet continuity can still be discussed quite sensibly.
Suppose X is some space, and that it's equipped with some topology Á. We say that X is Hausdorff (or T2) provided its points can be separated from one another by open sets. That is, if x and y are distinct points in X, we can find open sets U and V in Á with x Î U and y Î V, where U and V have no points in common. Being Hausdorff essentially means that there are enough open sets to stop the points in X getting 'tangled up' with one another.
If { Ui } is some collection of open sets whose union is the whole of X, we call it an open cover of X. Open covers can have very few members or very many members. For example, since X is itself open, the singleton {X} is itself an open cover. So is the entire topology, Á . Often, it turns out that some of the sets in an open cover are redundant. For example, { X , Æ } is also an open cover of X, but the empty set Æ is obviously irrelevant. We can remove it from the cover, and what's left still covers X. If we can remove sets from a cover and what's left behind still covers X, we call the smaller collection a subcover of the larger one.
A space is compact provided every open cover has a subcover containing only finitely many members (a finite subcover). Given this definition, it turns out that R and Q are not compact, but that every closed and bounded subset of R is compact. In particular, the unit interval I = [0,1] is compact. Compact spaces are very popular with topologists, because they're very easy to reason about.
The easiest way to do this is to choose K to be a compact space that contains X. Of course, we don't want to make K bigger than it has to be, so we generally try to arrange for X to be dense in K. That means simply that there are no open sets in K which aren't needed - every open set in K contains at least one point from X. For example, every set is dense in its closure. If you can find a compact space K which contains a dense copy of X, then K is a compactification of X. Normally, K (and hence its subspace X) is required to be Hausdorff as well.
This limits which spaces X can be treated in this way, but it's only a minor problem because just about all of the most useful spaces can be handled using this approach. For example, imagine picking up the open interval (0,1) and wrapping it in a circle. Because the endpoints are missing, we can't complete the task. But by adding just one extra point (at the join) we can close the loop and so create a compact space. This means that the circle is a compactification of (0,1). Notice by the way that [0,1] is also a compactification of (0,1). If a space has one compactification, it often has lots of them.
In fact, X has a Hausdorff compactification if and only if it is a Tychonov (T3½) space - this jargon means that whenever A is closed in X and x Î X\A, there exists a continuous real-valued function which is identically zero on A and takes the value 1 at x. Alternatively, suppose that K is any compact Hausdorff space, and that U Í K. The closure of U is itself compact, and contains U densely. Consequently a space has a compactification (and so is Tychonov) if and only if it can be identified with a subspace of a compact Hausdorff space.
If f is one of these functions, we can use it to define a new function g. If f (x) < 0, we'll define g (x) = 0, and if f (x) > 1, we define g (x) = 1. Otherwise we define g (x) º f (x). Now g is still continuous, and still separates a from A, but with one crucial difference. Whereas the range of f is R, that of g is the compact Hausdorff space I º [0,1].
From now on let's focus only on the continuous functions of the form g : X ® I. For every such g, let Ig be a copy of I, and let P be the product of all these copies of [0,1]. It is a highly useful fact that the product of compact Hausdorff spaces is itself compact Hausdorff, so in particular P must be compact Hausdorff. What's more, we can define a function F: X ® P by declaring the Ig'th co-ordinate of F(x) to equal g(x), and it turns out that F is actually an embedding of X as a subspace of P. Since P is compact Hausdorff, the closure of X in P must be a compactification of X. We call this the Stone-Cech compactification, bX.
I conjectured way back in 1986 that bQ(n) and bQ(n+2) are actually the same space - and that both are possibly the same space as bQ(n+1). As yet, however, I know of no proof of this conjecture, nor of any proof that it is wrong.
More amazing is the fact that we can replace the space I with any other compact Hausdorff space K, and this functional extension result remains valid. Any continuous f : X ® K extends to a continuous f b : bX ® K. Incidentally, this extension function is always uniquely defined (this is a simple consequence of the fact that X is dense in bX).
In particular, if K is any other compactification of X, the identity function i : X ® K embedding X in K extends to a function ib : bX ® K. This extension carries X onto itself and X* onto K\X. So here we see another crucial property of bX, and the reason why it's so much more important than any other compactification. Every compactification of X is a quotient space of bX. In a sense, then, bX is the "largest" possible compactification X can have, and we can classify all the compactifications of the space X by constructing the quotients of bX.