A Taste of Topology 2
Aug. 12th, 2009 03:20 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowNow let's look at some examples, strange and otherwise, and what makes them strange.
First, since it's been a while, let's recap. A topological space consists of a set X, together with a collection T of some of its subsets, designated as "open". X is called the underlying space and T the topology on X. T must satisfy the following conditions: X and the empty set are open; the intersection of any two open sets is open; the union of any collection of open sets is open. The underlying idea is that if a point p is in an open set, every point "close to p" is also in that set.
This is an extremely broad definition. It has to be; topology has many applications, and the spaces appropriate to some of those applications can be very strange. Nonetheless, in the core areas of the field, we're interested in more-or-less geometric objects, and specifically objects which can be realized as subsets of some real space - the line, the plane, or some multidimensional equivalent. Putting a topological space X into real space amounts to giving it coordinates - one if we're putting it into the line, two for the plane, and so on. So it's crucially important to come up with functions from X to the real numbers.
Not just any functions, though. They have to pay attention to the topology; that is, they must be continuous. If you've taken calculus, you've run into a version of this concept, using epsilons and deltas. The topological definition is actually, on the surface, much simpler, but when applied to functions from the reals to the reals emerges as the definition from calculus.
Let X and Y be two topological spaces, and f a function from X to Y. We say that f is continuous if the following is true: for every open subset V of Y, the set f*V = {x ∈ X : f(x) ∈ V} is an open subset of X. Again, think of the underlying idea: the set of points which f sends somewhere close to f(x) includes all the points close enough to x. The "somewhere close" is the epsilon of the traditional definition, and the "close enough" is the delta.
But wait: I haven't specified a topology on the reals. It's the one you'd expect: a subset of the reals is open if, whenever it contains a number r, there is some small open interval around r which is also contained in that subset. The traditional open intervals and open rays are open in this topology, and so are arbitrary unions of these.
Okay, now let's look at some examples. The simplest sort of topology, in some sense, is the "indiscrete" topology. Take any set - for definiteness, let's let X={a,b,c} - and decree that the only open sets are the empty set and X itself. (Those two have to be open; nothing says that anything else needs to be.) This is an "indiscrete space". These are rather strange; in particular, we can't fit them into real space at all. Specifically, any continuous function f from an indiscrete space to the reals must be constant.
The problem is that every open set which contains a also contains b and c. Let's make another assumption to prevent that.
Is T1 enough? Unfortunately, no. Here's an interesting example of a T1 space. A curve in the plane is "algebraic" if it's the graph of some polynomial equation. So, for example, straight lines, circles, and conic sections are algebraic, as are the graphs of polynomial functions. We define the "Zariski topology" on the plane by giving the closed sets rather than the open ones: the Zariski-closed sets are single points, algebraic curves, and finite unions of these. The plane, with the Zariski topology, is a T1 space, and it's very important in modern number theory. Unfortunately, once again there are no continuous functions from the Zariski plane to the reals, except for constants. The problem this time is that, if U and V are two nonempty Zariski-open sets, their intersection is also nonempty. If a function f sent p to the number r and q to a different number s, we could construct little intervals I and J around r and s that didn't overlap; but then f*I and f*J couldn't overlap either, which is impossible.
So we move on to a stronger assumption.
There are stronger conditions. A regular space is a space which satisfies T1 and also
The various T-axioms are also called "separation axioms"; T2 says that we can "separate points", T3 that we can separate points from closed sets, and T4 that we can separate closed sets from each other. Every normal space is regular; every regular space is Hausdorff; and every Hausdorff space is T1. The converse fails in each case; there are Hausdorff spaces which are not regular and regular spaces which are not normal. (Constructing them, and verifying these claims, is a lengthy process, and I'm not going to bother you with it.)
Note, though, that nothing here says anything directly about the existence of continuous functions to the real numbers. The failure of one or another of these axioms may put limits on what continuous functions are possible, but there is no visible guarantee that any interesting continuous functions are possible. Urysohn's lemma, which I'll turn to in the next of these posts, rather surprisingly shows that any normal space has lots of interesting continuous functions to the reals. It may take a post or two to explain just why that works; stay tuned.
(If I'm going too fast - if you want more explanation - please let me know!)
First, since it's been a while, let's recap. A topological space consists of a set X, together with a collection T of some of its subsets, designated as "open". X is called the underlying space and T the topology on X. T must satisfy the following conditions: X and the empty set are open; the intersection of any two open sets is open; the union of any collection of open sets is open. The underlying idea is that if a point p is in an open set, every point "close to p" is also in that set.
This is an extremely broad definition. It has to be; topology has many applications, and the spaces appropriate to some of those applications can be very strange. Nonetheless, in the core areas of the field, we're interested in more-or-less geometric objects, and specifically objects which can be realized as subsets of some real space - the line, the plane, or some multidimensional equivalent. Putting a topological space X into real space amounts to giving it coordinates - one if we're putting it into the line, two for the plane, and so on. So it's crucially important to come up with functions from X to the real numbers.
Not just any functions, though. They have to pay attention to the topology; that is, they must be continuous. If you've taken calculus, you've run into a version of this concept, using epsilons and deltas. The topological definition is actually, on the surface, much simpler, but when applied to functions from the reals to the reals emerges as the definition from calculus.
Let X and Y be two topological spaces, and f a function from X to Y. We say that f is continuous if the following is true: for every open subset V of Y, the set f*V = {x ∈ X : f(x) ∈ V} is an open subset of X. Again, think of the underlying idea: the set of points which f sends somewhere close to f(x) includes all the points close enough to x. The "somewhere close" is the epsilon of the traditional definition, and the "close enough" is the delta.
But wait: I haven't specified a topology on the reals. It's the one you'd expect: a subset of the reals is open if, whenever it contains a number r, there is some small open interval around r which is also contained in that subset. The traditional open intervals and open rays are open in this topology, and so are arbitrary unions of these.
Okay, now let's look at some examples. The simplest sort of topology, in some sense, is the "indiscrete" topology. Take any set - for definiteness, let's let X={a,b,c} - and decree that the only open sets are the empty set and X itself. (Those two have to be open; nothing says that anything else needs to be.) This is an "indiscrete space". These are rather strange; in particular, we can't fit them into real space at all. Specifically, any continuous function f from an indiscrete space to the reals must be constant.
Suppose f sends a to the real number r. Any small open interval I around r is open, and so f*I must be open. But the only possibilities are the empty set and all of X, and since f*I contains a, it can't be empty; it must be X. That is, f must also send b and c into I. But I could be as small as we want; ultimately, this implies that f(b) and f(c) must actually equal r.In some sense, the points of an indiscrete space are "stuck together"; we can't separate them with a real-valued function.
The problem is that every open set which contains a also contains b and c. Let's make another assumption to prevent that.
Axiom T1: if x and y are two distinct points of X, there must be an open set U which contains x but not y, and likewise the other way around.The T1 axiom is equivalent to saying that single points are closed sets. (Recall that a set is closed if its complement is open.) A topological space which satisfies T1 - a T1 space - is much better behaved than a general topological space.
Is T1 enough? Unfortunately, no. Here's an interesting example of a T1 space. A curve in the plane is "algebraic" if it's the graph of some polynomial equation. So, for example, straight lines, circles, and conic sections are algebraic, as are the graphs of polynomial functions. We define the "Zariski topology" on the plane by giving the closed sets rather than the open ones: the Zariski-closed sets are single points, algebraic curves, and finite unions of these. The plane, with the Zariski topology, is a T1 space, and it's very important in modern number theory. Unfortunately, once again there are no continuous functions from the Zariski plane to the reals, except for constants. The problem this time is that, if U and V are two nonempty Zariski-open sets, their intersection is also nonempty. If a function f sent p to the number r and q to a different number s, we could construct little intervals I and J around r and s that didn't overlap; but then f*I and f*J couldn't overlap either, which is impossible.
So we move on to a stronger assumption.
Axiom T2: if x and y are two distinct points in X, there must be an open set U around x and an open set V around y, such that U and V do not intersect.T2 spaces, also called Hausdorff spaces, are much better behaved even than T1 spaces. The Zariski plane is T1 but not Hausdorff.
There are stronger conditions. A regular space is a space which satisfies T1 and also
Axiom T3: if x is a point of X and F is a closed subset of X which does not contain X, then there are open sets U and V containing x and F respectively, which do not intersect.A space is normal if it satisfies T1 and
Axiom T4: if F and G are two closed subsets of X which do not overlap, then there are open sets U and V containing F and G respectively, and which do not overlap.
The various T-axioms are also called "separation axioms"; T2 says that we can "separate points", T3 that we can separate points from closed sets, and T4 that we can separate closed sets from each other. Every normal space is regular; every regular space is Hausdorff; and every Hausdorff space is T1. The converse fails in each case; there are Hausdorff spaces which are not regular and regular spaces which are not normal. (Constructing them, and verifying these claims, is a lengthy process, and I'm not going to bother you with it.)
Note, though, that nothing here says anything directly about the existence of continuous functions to the real numbers. The failure of one or another of these axioms may put limits on what continuous functions are possible, but there is no visible guarantee that any interesting continuous functions are possible. Urysohn's lemma, which I'll turn to in the next of these posts, rather surprisingly shows that any normal space has lots of interesting continuous functions to the reals. It may take a post or two to explain just why that works; stay tuned.
(If I'm going too fast - if you want more explanation - please let me know!)
