Proposition S1.32. If { Y α } is a family of topological spaces, each of which contains more than one point, and Π Y α. is metrizable, then each

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1 Proposition S1.32. If { Y α } is a family of topological spaces, each of which contains more than one point, and Π Y α. is metrizable, then each factor Y α is metrizable. (Reference: Dugundji)

2 Definition S1.28. The diameter of a metric space is defined as: diam(x, ρ) = sup x,y (ρ(x,y)). If we wish, we can rescale a metric as in Exercise S1.25 so that the diameter of X is finite, indeed as small as we wish. Exercise S1.29. Show that if {ρ α } is a family of pseudometrics on X, such that diam(x, ρ) M < for all α, then ρ sup (x,y) := sup α (ρ α (x,y)) is a pseudometric on X. Exercise S1.30. Show that if {ρ n } is a finite or countable family of pseudometrics on X, then ρ (x,y) := n (ρ n (x,y)) is a pseudometric on X, provided that it converges for all x,y X. If {ρ α } is a family of pseudometrics on spaces {Y a }, then each one of them also works as a pseudometric on the Cartesian product Π Y α. To be precise, a pseudometric on Π Y α can be defined by ρ α,# (x,y) := ρ α ( π α (x), π α (y)). It is straightforward to see that this satisfies all the conditions of a metric. Proposition S1.31. A Cartesian product of a countably infinite family of metric spaces is metrizable. A metric on it is given by ρ sup (x,y) where the metrics {ρ n } on the factors have been scaled so the diameters are all bounded by a finite number M. This proposition does not guarantee that the metric topology is the same as the product topology. A necessary and sufficient condition for this is that lim diam(x n, ρ n ) = 0.

3 Theorem S1.24 (Urysohn). Assuming that the topology of X has a countable base, X is metrizable iff T3. (Reference: Dugundji.) Suppose we do not want points in some space X to be too far apart, which they appear to be according to some metric ρ. E.g., we don't like the fact that points in R 2 can be arbitrarily far apart. We can quantify this situation by defining the diameter of a space as sup(ρ(x,y): x,y X), and worrying that the diameter of R 2 is infinite. Well, we can fix that in two plausible ways. First, we could use the tangent function to create a homeomorphism between R and I and using the pull-back trick to make R 2 look like I 2. This is somewhat particular to the example of products of R, however. A more universal method is to the original metric with one defined by ρ fin (x,y) := ρ(x,y)/(1 + ρ(x,y)) Exercise S1.25: Show that if ρ is a (pseudo)metric on X, then so is D ρ fin for any number D>0, and the diameter of X in the latter (pseudo)metric is at most D. Definition S1.26. Two (pseudo)metrics are topologically equivalent iff their metric topologies are the same. Exercise S1.27: Show that ρ and D ρ fin are topologically equivalent. There are two other useful ways to generate new metrics, which are not generally equivalent to any of the metrics they are built from. Indeed, it frequently happens that pseudometrics can be used to produce a true metric by thesee tricks:

4 Note: We do not always wish to give a Cartesian product the product topology. The closed unit ball in Hilbert space, given in Example S1.11 is compact in the weak topology, because it is a closed subset of the Cartesian product of an infinite number of sets of the form x n 1, but as we have seen, it is not compact in the more usual norm topology. It is a difficulty in analysis that the more usual topology on the this example does not allow it to be compact. Example S1.22. To see the difference in a striking way, recall the sequence of points e k := {δ k,n } in Example S1.11. This sequence converges to zero in the product topology (usually called the weak Hilbert-space topology), but it does not converge to 0 in norm, since e k = 1 for all k. More about metrics and pseudometrics. It is often useful to cook up new metrics, or pseudo metrics, from old ones with simple tricks. One way to do this is to use a homeomorphism between two spaces, h: X Y, where one of the spaces has a convenient (pseudo)metric ρ, say Y carries the (pseudo)metric ρ. Then, as is easy to verify, the pull-back (pseudo)metric ρ (x 1,x 2 ) := ρ(h(x 1 ),h(x 2 )) is a (pseudo)metric defined on X. Structures preserved by homeomorphisms are called topological invariants. Definition S1.23. A topological space X is metrizable iff it is possible to define a metric on X such that its topology is the metric topology. According to the preceding paragraph, metrizability is a topological invariant.

5 then it is also Hausdorff with respect to any finer topology T 1 T. A similar statement applies to the other separation axioms. The product topology is the roughest topology on Π Y α such that convergence of a sequence implies convergence of the projection of the sequence onto any factor Y α. Hence convergence in any other topology with this property implies convergence in the weak topology. Theorem S1.21 (Tychonov). A Cartesian product of topological spaces, given the product topology, is compact iff each factor is compact. Proof that each factor is compact if the product is compact. The projection map π α is by construction continuous from the product Π to the factor X α, and we know that continuous images of compact spaces are compact. Proof of the converse, under the added assumption that Π Y α is a countable product of countably compact spaces. Let {x (k) } be a sequence in Π Y n. That is, for each k = 1, 2,..., x (k) is of the form x (k) = {x n (k) } If Y 1 is compact, there is a subsequence k=k1 beginning with k1=1, such that x 1 (k1) converges as k1. We may assume that If Y 2 is compact, there is a subsubsequence k1=k2, beginning with k2=1 and then k2= the second value of k1, such that both x 1 (k2) and x 2 (k2) converge. Continuing, we produce sub...subsequences such that x 1 (kn)... x n (kn) converge for all n, where kn = {1, the second value of k1, the third value of k2,..., the m+1st value of km...} for all m < n. In this way we construct a subsequence {x (kj) }, j= 1,2,..., which converges in the product topology.

6 an angular coordinate usually called θ or φ, identifying 0 with 2π in the usual way. R is the usual Euclidean line, and I is the open unit interval. I and R are homeomorphic (by the map y = tan(2x/π), so are interchangeable to a topologist. Examples S1.19. R 2 = R R. The usual Cartesian coordinate representation corresponds to this. I I is the unit square. T 2 = T 1 T 1. This is the familiar (2 dimensional) torus. T n is referred to as the n-torus. Since S 1 = T 1, obviously the notation S n for the n-sphere (the 2-sphere being ths usual surface in R 3 ) does not follow the same pattern for Cartesian products. S 1 R is a cylinder. (S 1 I would be a cylinder of finite height, but that would require putting the usual metric on I, and we haven't spoken of metrics in this section yet). We can create the 2-sphere by appending a north and south pole as with the 2-point compactification of the line: S 1 R {N} {S}, or as a one-point compactification of the plane, R 2 {N}, using the stereographic projection to identify the north pole with the point at infinity. Y X := Π X Y is the set of all functions from X to Y, with no conditions of regularity at all. Proposition S1.20. A Cartesian product of Hausdorff spaces given the product topology is Hausdorff. For this proposition, we implicitly assumed that the product is given the product topology. What is we give it some other topology? If a space is Hausdorff with respect to some topology T,

7 Exercise S1.12. In a pseudometric space, total boundedness implies boundedness. Exercise S1.13. In R n, total boundedness and boundedness are equivalent. Lemma S1.14. Every countably compact pseudometric space is totally bounded. Lemma S1.15. A countably compact pseudometric space is compact. Finally, with a proof patterned on that of the Heine-Borel Theorem, we can conclude: Theorem S1.16. A complete pseudometric space is compact iff it is closed and totally bounded. Example S1.17: Hilbert cube. The Hilbert cube is the subset of the unit ball in the preceding example such that x n 1/2 n. (Some books define it with 1/n, but then it is not a subset of the unit ball) Exercise S1.18. Show that the Hilbert cube is compact in the topology given by the norm {x n } 2 by using the compactness of closed bounded sets in R N, which is "isomorphic" (identical in structure) to the set of sequences for which x n =0 when n>n, and using the triangle inequality to take care of possibly nonzero x n for n>n. Supplementum to Cain Chapter 5, on product spaces. Here are some familiar examples of spaces expressed as Cartesian products. Here S 1 = T 1 both denote a circle, for which we can use

8 radii. The most familiar example is R n, as the set of points with rational coordinates is dense. Definition S1.9. A space with a countable dense subset is called separable. Recall that in R n, a set is compact iff it is closed and bounded. This is known as the Heine-Borel Theorem. It needs to be modified for general pseudometric spaces. A key notion is total boundedness, defined in Cain, above Proposition 6.4. In class when we defined total boundedness we introduced the term "εnet". Definition S1.10. An ε-net is a covering of a set by cells (open balls) of fixed radius ε. A set S is totally bounded iff for every ε>0, S has a finite ε-net. Total boundedness is clearly implied by compactness, and it is easier to check. The important fact about complete metric spaces is that they are compact iff they are closed and totally bounded. Boundedness is not equivalent to total boundedness, as shown by the following example. Example S1.11: The unit ball in Hilbert space. Let l 2 denote the space of all sequences {x n }of complex numbers such that Σ x n 2 < }. (Observe that this infinite-dimensional space is a separable metric space.) The closed unit ball { {x n }: Σ x n 2 1} is not compact in the topology given by the norm {x n } 2. For the sequences e k := {δ k,n }, where δ k,n := 0 when k n and := 1 when k=n, are a fixed distance apart: e j -e k = 2 when j k. Since the "points" {e k } cannot be covered by a finite number of balls of radius < 1/ 2, neither can the unit ball.

9 T4. Given any pair of closed set A 1,2 X, there exist disjoint open sets U 1,2 such that A 1,2 U 1,2. A space is normal if it satisfies axiom T4. Theorem S1.5 (Urysohn). A space is normal iff for every pair of closed sets A,B there exists a continuous function f A,B, such that a A f A,B (a) = 0, and b B f A,B (b) = 1. Such a function is called an Urysohn function. The main point of compactness, for an analyst, is that sequences in compact sets have cluster points, and hence convergent subsequences. Corollary S1.6 of Cain's Theorem 4.8. Every infinite subset of a complete compact space has a cluster point. In particular, every sequence in a compact subset K of a topological space X has a subsequence that converges in K. This is a version of Cain's Theorem Euclidean spaces and most of the other spaces we will encounter in this class have some special structures which make analysis easier. One of these properties is the fact that they are countably compact. This allows us to discuss all convergence questions in terms of countable sets, that is, sequences. Definition S1.7. A space or set is countably compact iff every countable open covering has a finite subcovering. Proposition S1.8. If a topological space has a countable base, then compactness and countable compactness are equivalent. For instance, if there is a sequence in a pseudometric space X which is dense in X, then the topology has a countable base, composed of open balls centered at the points of the sequence, with rational

10 Proposition S1.3. If f is continuous and C is connected, then f(c) is connected. Corollary S1.4 (the intermediate value theorem). If f is a continuous function on a connected set C, taking values in R, and f(a) < f(b), then for all z, f(a) < z < f(b), these exists c C such that f(c)= z. Supplementum to Cain Chapters 4 and 6, on compact spaces and sequences. Hausdorff and others investigated the structure of topological spaces by asking how possible it is to separate points and subsets by covering them with open sets. The Axioms of separation. There are four successively stronger axioms of separation that a topological space might satisfy: T1. Given any pair of points x 1,x 2 X, there exist open sets U 1,2 such that x k U k, but x 1 U 2, and x 2 U 1. T2. Given any pair of points x 1,x 2 X, there exist disjoint open sets U 1,2 such that x k U k. A space is Hausdorff if it satisfies axiom T2. T3. Given any pair of point x X, and any closed set A X with x A, there exist disjoint open sets U 1,2 such that x U 1, A U 2. A space is regular if it satisfies axiom T3.

11 Supplementary lecture notes in Mathematics 6021A, February, Supplementum to Cain Chapters 1 and 2. Often we don't specify a topology with a clearly defined base, but as the "smallest" topology containing a given collection of subsets. The set that defines the topology is called a subbase. While a subbase is not as useful as a base, it will often suffice for our purposes. The smallest topology containing a family of subsets is welldefined because the intersection of topologies is always a topology. You can see this directly from the definition of a topology. (Notice, by the way, that it is not automatic that the union of topologies is a topology.) Since any collection of subsets of X belongs to the discrete topology on X, it is not vacuous when we define the smallest topology containing as the intersection of all topologies containing. As an example, the weak topology on X by a family of functions f α : X Y α is the one generated by the subbase {f α -1 (U α,β ): U α,β T(Y α )}. Supplementum to Cain Chapter 3, on connected spaces. Example S1.1. If S is a nonempty connected subset of R, then S is either a single point {x 0 } or an interval. Simple exercise S1.2. Show that the intersection of two connected sets need not be connected.