Metric and metrizable spaces These notes discuss the same topic as Sections 20 and 2 of Munkres book; some notions (Symmetric, -metric, Ψ-spaces...) are not discussed in Munkres book.. Symmetric, -metric, and metric Definition. Let X be a set. A function d : X 2 R is called a symmetric on X if d satisfies the following two axioms: () ( x, y X) d(x, y) 0 and d(x, y) = 0 iff x = y; (2) ( x, y X) d(x, y) = d(y, x) (the axiom of symmetry. If, in addition, d satisfies the following axiom of triangle (3) ( x, y, z X) d(x, z) d(x, y) + d(y, z), then d is called a metric on X. A function d that satisfies axioms () and (3) is called a -metric. If d is a metric (symmetric, -metric) on a set X then the pair (X, d) is called a metric (resp. symmetric, -metric) space. Examples 2. () X = R; d(x, y) = x y is the standard metric on R; n (2) X = R n ; d(x, y) = (x(k) y(k)) 2 is the Euclidean metric on R n ; (3) X = R n ; d(x, y) = n x(k) y(k) is the NY-metric on R n ; (4) X = R n ; d(x, y) = max k n x(k) y(k) is the maximum metric on R n ; (5) Consider the set X from Example 22 in the notes Countability conditions and convergence. For x, y X, put d(x, y) = d(y, x) = m if one of x, y is b m and the other is c; n if one of x, y is a m,n and the other is b m ; in all remaining cases. Then d is a symmetric on X which does not satisfy axiom 3 and thus is not a metric; (6) X is any non empty set; { 0 if x = y; d(x, y) = if x y is a metric on X. Definition 3. Let X be a set, x X, d a (sym)metric on X and ε > 0. The ε-ball around x is O ε (x) = {y X : d(x, y) < ε}. Exercise 4. Sketch ε-balls for Examples 2; in cases 2, 3, 4, take n = 2. Definition 5. A topology T on the set X is said to be generated by the metric d on X if the family {O ε (x) : x X, ε > 0} is a base for T. If a topology T on a set X can be generated by some metric on X then we say that the space (X, T ) is metrizable. A topology T on the set X is said to be generated by the symmetric d on X if the family if the following two conditions are equivalent for U X: Here and in the next two examples, n N, x = x(),..., x(n), y = y(),..., y(n).
2 U T ; for every x U there is ε > 0 such that O ε (x) U. If a topology T on a set X can be generated by some symmetric on X then we say that the space (X, T ) is symmetrizable. 2 Examples 6. The metric d from Example 2(6) generates the discrete topology on X. The metric d from Example 2() generates the standard topology on R. The metrics d from Examples 2(2,3,4) all generate the standard topology on R n. The semymetric d from Example 2(5) generates the topology on X that we discussed in Example 22 in the notes Countability conditions and convergence, and then mentioned again when discussing quotient spaces and quotient mappings. Recall that this topology is not first-countable. 2. Properties of metrizable spaces; more on countability conditions Proposition 7. Every metizable space is Hausdorff. 3 Proposition 8. Every metizable space is first countable. Exercise 9. Give an example of a metrizable space which is not second countable. Definition 0. Let (X, T ) be a topological space. A subset D X is called dense in X if D = X. If X has a dense countable subset then X is called separable. Definition. Let X be a set. A family U of subsets of X is called a cover of X 4 if U = X. A space (X, T ) is called Lindelöf if for every cover U of X consisting of open sets 5 there exists a countable subfamily of U that still covers X. Theorem 2. Every second countable space is both separable and Lindelöf. Examples 3. () The Sorgenfrey line is both separable and Lindelöf but (as we already know) not second countable. (2) The one-point Lindelöfication of a discrete space. Let D be an uncountable set, and p D. Put X = D {p}. Topologize X declaring the points of D isolated while a basic neighborhood of p takes the form O C = {p} (D\C) where C is an arbitrary countable subset of D. Then X is Lindelöf but non separable and thus non second countable. (3) (R, T fin ) is separable but non Lindelöf and thus non second countable. Theorem 4. For a metrizable space X, the following conditions are equivalent: () X is second countable; (2) X is separable; (3) X is Lindelöf. 6 2 The reason for the difference in the definitions is that, in the case of a metric d, the balls O ε(x) have to be open in the topology generated by d while in the case of a symmetric d they do not have to. 3 In fact, much better than Hausdorff, we will see this later 4 Or we say that U covers X 5 briefly: for every open cover 6 The proof will be discussed in class.
3 Proposition 5. Every subspace of a metrizable space is metrizable. Proposition 6. Every finite or countable 7 product of metrizable spaces is metrizable. A product of uncountably many factors each of which consists of more than one point is (not first countable therefore) non metrizable. 3. Properties (and more examples) of symmetrizable spaces Proposition 7. Every symmetrizable space is T. Proposition 8. Every symmetrizable space is sequential. Examples 9. () Niemytzky plane (see Hw 3, Part, Problem ) is symmetrizable. (2) Let s return to Example 2, (5). This space is symmetrizable, hence sequential, but, as we already know, it is not Fréchet. So a symmetrizable space 4. Just a little about -metrizable spaces The Sorgenfrey line is -metrizable 8 but not symmetrizable. 9 A real life example of a -metric space which is not symmetric: in a city, measure the distance between points by the time needed to travel; consider a city with some one-way routes. Another example: traveling in the mountains. A real life example of a symmetric space which is not -metric: measure the distance between cities by the time needed to travel by air. Assume that if there is a flight from A to B, then there is one from B to A that takes the same time. Now suppose there is no direct flight from A to C, but there is one from A to B, and one from B to C, and at B you have to wait. 0 5. One more generalization of metric spaces; Ψ-spaces Definition 20. A function d : X 2 R that satisfies axiom from Definition is called a o-metric on X. Then the pair (X, d) is called an o-metric space. ε-balls are defined like in Definition 3. If the topology T on X is generated by the o-metric d in the way described in Definition 5 for symmetrics, then the pair (X, T ) is called an o-metrizable (or weakly first-countable ) space. Example 2. The Alexandroff-Urysohn Ψ-space. For each r P, fix a subset S r of Q which can be enumerated as a sequence converging to r. Consider the topology T Ψ on R such that the points of Q are isolated in T Ψ while a basic neighborhood of a point r P takes the form {r} (S r \ F ) where F is an arbitrary finite subset of S r. Let us call the pair (R, T Ψ ) an AU-Ψ-space. 7 Countable products will be discussed later 8 Very easy to see 9 The proof that I know uses the Baire property (discussed in Chapter 8 of Munkres book). 0 This section is optional Guess why
4 Let E denote the standard topology on R. Then T Ψ E, so the identity mapping i : R R is a condensation of an AU-Ψ-space onto the real line with the standard topology. Note that Q is an open discrete subspace of (R, T Ψ ), and P is a closed discrete subspace of (R, T Ψ ). (R, T Ψ ) is first-countable, hence o-mtrizable. With an accurate choice of the sequences S r, (R, T Ψ ) can be made symmetrizable. Since (R, T Ψ ) contains an uncountable discrete subspace, (R, T Ψ ) is not second countable. Moreover, since (R, T Ψ ) contains an uncountable closed discrete subspace, (R, T Ψ ) is not even Lindelöf. 2 Since (R, T Ψ ) is separable but not second countable, (R, T Ψ ) is not metrizable. Example 22. The general definition of a Ψ-space. Let N be a countably infinite set. A family R of countably infinite subsets of N is called almost disjoint if for every two distinct r, r 2 R, the intersection r r 2 is finite. 3 So let R be an almost disjoint family of infinite subsets of N. The set Ψ(R) = N R is topologized by declaring the points of N isolated while a basic neighborhood of r R takes the form {r} (r \ F ) where F is an arbitrary finite subset of r. 4 A general Ψ space has properties similar to those of the AU-Ψ-space: N is open and discrete in Ψ(R) while R is closed and discrete. Ψ(R) is always first countable. Ψ(R) is second countable iff R is countable (and in this case Ψ(R) is homeomorphic to the discrete union of countably many copies of the space convergent sequence ). Definition 23. A function f : X R is bounded if there is a constant M such that f(x) < M for every x X. A topological space is pseudocompact if every continuous function f : X R is bounded. Definition 24. Let N be a countably infinite set. An almost disjoint family R consisting of infinite subsets of N is called maximal almost disjoint 5 if R is not a proper subset of any larger almost disjoint family consisting of infinite subsets of N. One can prove using the axiom of Choice that every almost disjoint family can be extended to a maximal almost disjoint family. Proposition 25. Ψ(R) is pseudocompact iff R is a mad family. Ψ(R) with a mad family R is not symmetrizable except when R is finite. 6. Bounded metrics It is natural to call a metric d on a set X bounded if there is a constant M such that d(x, y) < M for every x, y X. Notation 26. Let d be a metric on a set X. For every x, y X, put d(x, y) = min{d(x, y), }. Then d is called the standard bounded metric corresponding to d. 2 It is a good exercise to show that a closed discrete subspace of a Lindelöf space can be at most countable. 3 Thus the family {Sr : r P} in Example 2 is almost disjoint because sequences converging to different points may have only finitely many common points. This also shows that an almost disjoint family of subsets of a countable set may have cardinality as big as c (while a disjoint family of subsets of a countable space is at most contable). 4 So r R is, at the same time, a point of Ψ(R), and a subset of Ψ(R). 5 Abbreviated as mad
5 Proposition 27. () d is a metric. (2) The topology on X generated by d is the same as the topology generated by d. So every metrizable topology can be generated by a bounded metric. Here is another way: Exercise 28. Let d be a metric on a set X. For every x, y X, put d(x, y) = Arctan d(x, y). Show that d is a metric on X, and that d generates the same topology on X as d. 7. Metrics on finite products Examples 29. Let n N and for each k {,..., n} let (X k, T k ) be a topological space and d k a metric on X k that generates the topology T k. For x = x(),..., x(n), y = y(),..., y(n) P = X i put d NY (x, y) = i n d Euclidean (x, y) = n (x(k) y(k)) 2 ; n x(k) y(k) ; d max (x, y) = max x(k) y(k). k n Then each of d Euclidean, d NY, d max (is a metric and) generates the Tychonoff product topology on P. 8. Metrics on infinite products Definition 30. Let for each a A, (X a, T a ) be a topological space such that the topology T a is generated by the metric d a. For x, y P = X a, put d(x, y) = sup d a (x(a), y(a)). a A Then d is called the uniform metric on P and the topology generated by d is called the uniform topology. The uniform topology on P is stronger than the Tychonoff product topology but weaker than the box product topology. Proposition 3. Let for each n N, (X n, T n ) be a topological space such that the topology T n is generated by the metric d n. For x, y P = X n, put D(x, y) = sup n N n d n(x(n), y(n)). Then D generates the Tychonoff product topology on P. 9. Mappings of metric spaces Proposition 32. Let (X, d X ) and (Y, d Y ) be two metric spaces. A mapping f : X Y is continuous (with respect to the topologies generated on X and Y by d X and d Y ) iff for every x 0 X and every ε > 0 there exists δ > 0 such that for every x X, if d X (x, x 0 ) < δ then d Y (f(x), f(x 0 )) < ε. a A n N
6 Definition 33. Let (X, d X ) and (Y, d Y ) be two metric spaces. A mapping f : X Y is called uniformly continuous (with respect to the metrics d X and d Y ) if for every ε > 0 there exists δ > 0 such that for every x 0, x X, if d X (x, x 0 ) < δ then d Y (f(x), f(x 0 )) < ε. Example 34. The mapping f : R R is continuous (with respect to the standard topology on R) but not uniformly continuous (with respect to the standard metric on R. Definition 35. Let (X, T ) be a topological space. A sequence of functions {f n : n N} (where f n : X R) converges pointwise to a function f : X R if for every x X, lim f n(x) = f(x) (that is for every x X for every ε > 0 there is n N(ε, x) such that for all n > N(ε, x), f n (x) f(x) < ε). In this case we say that f is the pointwise limit of the sequence {f n : n N}. A sequence of functions {f n : n N} (where f n : X R) converges uniformly to a function f : X R if for every ε > 0 there is N(ε) such that for all n > N(ε), for every x X, f n (x) f(x) < ε).in this case we say that f is the uniform limit of the sequence {f n : n N}. Definition 35 corresponds to the convergence in the sense of pointwise or uniform topology on R X. 6 Example 36. The pointwise limit of a sequence of continuous functions is not necessarily continuous. Indeed, put 0 if x 0; f n (x) = nx if 0 x n ; if x n for n N and f(x) = { 0 if x 0; if x > 0. Proposition 37. The uniform limit of a sequence of continuous functions is a continuous function. 6 Note that typically R X is not sequential, so convergent are not enough to describe its topology.