Notes on Metric Spaces

MATH MA TH 104—INTRODUCTION 104—INTRODUCTION TO ANALYSIS ANALYSIS SPRING SPRING 2012 NOTES NOTES ON METRIC METRIC SPACE SPACES S RUMEN ZAREV

Contents

1. Mo Moti tiv vati ation on 1.1.. So 1.1 Solu lutio tion n 2. Me Metr tric ic sp spac aces es 2.1. Dis Distan tance ce fun functio ctions ns on Rn 2.2. Dis Distan tance ce funct function ionss on the circ circle le 2.3. Un Unifor iform m metric metric on the space space of func function tionss 3. Con Conve verge rgence nce in in metric metric spac spaces es 4. Topolo opologic gical al spa spaces ces 5. Clos Closed ed and com compac pactt sets sets 5.1. Equ Equiv ivale alence nce of metr metrics ics 5.2. Mor Moree topologic topological al notion notionss in metric metric spaces spaces 6. Non Non-to -topolog pological ical aspect aspectss of metric metric spaces spaces

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Motivation

When talking about notions such as convergence, continuity, etc. the key notion is that of points (numbers) being “close to each other”, or “arbitrarily close”. close”. For example when working working with the extended real numbers we have encountered the following situations: • We say that the real numbers x and y are close to each other if |x − y | is a small positive number. We say that we can find “xn arbitrarily close to y ” if we can find xn such that |xn − y| < ε for any ε > 0. • We say that the real number x is close to +∞, if  x is a large positive number. We say that we can find “xn arbitrarily close to +∞” if we can find xn > M  for any M  ∈ R. We can use this language to phrase some familiar notions, like the limit of  a sequence of real numbers: lim xn = x if for large n, xn is arbitrarily close to x. This This would work work both for the case of  x being a real number, and for the cases of  x being ±∞. Similarly, for a function f  : R → R we can define limx→a f (x) = L as follows: “for x close to a, f (x) becomes arbitrarily arbitrarily close to L”. 1

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Unfortunately, as we have seen, if we want to use a definition like this in practice we have to consider multiple cases. Ignoring left and right limits, there are nine distinct cases, depending on whether a is real, −∞, or +∞, and similarly, whether L is real, −∞, or +∞. This means that really we have nine different definitions of limit of a function. Not only that, but if we want to prove or use any theorem about limits, we have to consider all these cases separately—we would have to prove nine different theorems, even when they are all very similar. What if we want to work with convergence, limits, or continuity (and even derivatives, and integrals) in a setting other than the real numbers? What if we are considering multidimensional quantities, e.g. points in a plane, or in space? What about sequences of functions? We have already seen two notions of convergence of sequences of functions—pointwise and uniform. Which one works better? Why? 1.1. Solution. As is standard in mathematics, we can solve all of these problems together, by developing a general abstract theory. That is, take the intuitive notion we are using—“closeness”, and make it precise. If we can come up with a rigorous definition that captures this, then we can work in full generality. Every theorem we prove in the general case translates into many different results in the specific cases. In fact, we will consider two separate constructions: • Topological spaces: In a topological space the fundamental notion is that of an open set . It captures the qualitative part of “closeness”. It makes the statement “arbitrarily close” precise. • Metric spaces: In a metric space the fundamental notion is that of  distance . It captures the quantitative part of “closeness”. In addition to “arbitrarily close”, we can make also makes statements like “x is closer to y than z ” precise. Of the two, metric spaces are more concrete, and more similar to the real numbers. Thus they are easier to understand, and we will deal with them first. 2.

Metric spaces

A metric space consists of some objects (e.g. numbers, points) and a notion of distance between pairs of objects. We can make that precise as follows. Definition 1. A metric space consists of a pair of  (X, d), where  X  is a set, called the  space, and  d : X  × X  → R is a function, called the  metric or  distance function, assigning a real number to pairs of elements of  X . This 

distance function satisfies the following conditions: (M1 ) d(x, y) = d(y, x) for all  x, y ∈ X . (M2 ) d(x, x) = 0 for all  x ∈ X , and  d(x, y) > 0 whenever  x  = y. (M3 ) d(x, y) + d(y, z ) ≥ d(x, z ) for all  x , y , z ∈ X .

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All three of these conditions make sense, if we think of  d as distance. (M1 ) means that d is symmetric—the distance from x to y is the same as that from y to x. (M2 ) means that the distance from x to itself is always 0, while the distance between any two distinct points is strictly positive. Finally, (M3 ) is the triangle inequality —going from x to y to z , cannot be shorter that going from x to z directly. For a “real world” example of a metric space consider the following. Suppose that X  is the set of cities in a country, connected by a highway system, e.g. the United Sates. Let d(x, y) be thew shortest distance (in miles), along any route, between the cities x and y, e.g. we can have d(Berkeley, New York) = 2897, d(Berkeley, Los Angeles) = 2777, and d(Berkeley, Los Angeles) = 374. As we see, the triangle inequality is satisfied for these three cities. For a more mathematical example, notice that we have already seen the properties (M1 )–(M3 ). They are satisfied by the absolute value function. This gives us our first mathematical example. Example  2. We can turn the set of real numbers into a metric space (R, dstd ) by using the standard distance function  dstd (x, y) = |x − y|. Indeed, this is the usual notion of distance between two real numbers, and coincides with the distance between the points on the real line representing the numbers. For instance d(1, 4) = 3, and d(−5, 6) = 11. 2.1. Distance functions on Rn . Another familiar example of a distance function is that between points in the plane. If we put a coordinate system, and using the Pythagorean theorem, between   the usual2 notion of a distance the points (x1 , x2 ) and (y1 , y2 ) is (x1 − y1 ) + (x2 − y2 )2 . In general, for any natural number n, we can turn the n–dimensional space Rn into a metric space. Example  3. For any n ≥ 1, there is a metric space (Rn , d2 ), where the distance between two points x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) is d2 (x, y) =

 

(x1 − y1 )2 + · · · + (xn − yn )2 .

It is easy to see that this function satisfies (M1 ) and (M2 ). It is a little harder to see it satisfies the triangle inequality (M3 ). There are two other important metrics that can be put on Rn . One of  them is called the taxi-cab metric . To understand it imagine the following situation—we are considering the points in the plane R2 as addresses in a city. To get from one address to another we cannot travel along a straight line as there are buildings in the way. Instead, we have to travel along a grid-like street system—we can only move in a North–South or an East– West direction. The distance between two points is then the sum of the NS distance and the EW distance between them. More formally we have the following.

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Example  4. For any n ≥ 1, there is a metric space (Rn , d1 ), where the distance between the points x = (x1 , . . . , xn ) and y = ( y1 , . . . , yn ) is d1 (x, y) = |x1 − y1 | + · · · + |xn − yn |.

The third metric we would like to consider is called supremum  or uni form  metric. In this case the distance between two points is the maximum difference in any coordinate. Example  5. For any n ≥ 1, there is a metric space (Rn , d∞ ), where the distance between the points x = (x1 , . . . , xn ) and y = ( y1 , . . . , yn ) is d∞ (x, y) = max{|x1 − y1 |, . . . , |xn − yn |}.

Notice that we have used very similar notation for two of these metrics— d1 and d2 . This is due to the fact that they are special cases of a more general construction, called the p–metric. Example  6. For any n ≥ 1 and real number p ≥ 1, there is a metric space (Rn , d p ), where the distance between the points x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) is d p (x, y) =

  p

|x1 − y1 | p + · · · + |xn − yn | p .

We can try do that for any positive number p. However, unless p is at least 1, the condition (M3 ) fails. The reason the supremum metric is denoted by d∞ is that it is, in a sense, the limit of d p as p goes to infinity. More precisely, for any fixed pair x, y ∈ Rn , we have d∞ (x, y) = lim p→+∞ d p (x, y). Exercise  7. For a fixed natural number n check that d p for p ∈ [1, +∞) and d∞ are indeed metrics on Rn , that is they satisfy the three conditions (M1 )–(M3 ). Hint: for the triangle inequality (which is the hardest to show) first try the case y = (0, . . . , 0). The general case is very similar. Notice for R1 = R, i.e. the real numbers, all of these metrics coincide (and are equal to the standard metric dstd ). To get a grasp of how these distance functions behave, it helps to visualize a “sphere”, i.e. the set of points equidistant to some fixed point. For instance, in R2 we can visualize the “sphere” of radius 1 around (0, 0) according to different metrics. For the Euclidean metric d2 we get the usual circle. However, for other metrics we get different pictures—squares for d1 and d∞ , and something between a square and a circle for say d3 . This can be seen in Fig. 1. The points (0, 1), (0, −1), (1, 0), and (−1, 0) are included in all four cases, but the rest of the points differ. 2.2. Distance functions on the circle. Consider the standard circle S 1 = {(x, y) : x2 + y2 = 1}. Instead of by Euclidean coordinates, we can refer to points on the circle by their angle with the x–axis. That is, we consider points θ ∈ [0, 2π), where θ corresponds to the point in R2 with coordinates (x(θ), y (θ)) = (cos θ, sin θ) (see Fig. 2). We can put a distance function on the circle by measuring the Euclidean distance in R2 , that is the distance along a straight line.

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Points in the “sphere” {x ∈ R2 : d(x, (0, 0)) = 1} for different metrics d. From left to right we have d1 , d3 , d2 , and d . Figure 1.

∞

Figure 2.

The correspondence between θ ∈ [0, 2π) and a point ( x, y ) on the

1

circle S  .

Example  8. There is a metric space ([0, 2π ), d2 ), where d2 (θ, ϕ) =

  θ − ϕ . (cos θ − cos ϕ)2 + (sin θ − sin ϕ)2 = 2 sin  2 

 

There are two ways to check that d2 is a metric on [0, 2π). One is to do it directly, using the formula. The other is by using the fact that we know it is a metric on the bigger space R2 , which contains S 1 . This is an example of the subspace  construction. Definition 9. Suppose  (X, dX ) is a metric space, and  Y  is any subset of  Y  . We can turn into a metric space by putting on it the  subspace metric dY   : Y  × Y  → R, which is just the restriction  dY   = ( dX ) |Y  ×Y  . Theorem 10. For any metric space  (X, dX ) and any  Y  ⊂ X , and for  dY   as in Def. 9 , (Y, dY  ) is a metric space.

Proof. The proof consists of checking that dY   satisfies (M1 )–(M3 ). This is straightforward from the fact that dX  satisfies them.  This metric on [0, 2π ) is somewhat hard to work with as it involves trigonometric functions. Instead, we can measure distances along the circle, instead of along a straight line (see Fig. 3). This distance has a much easier formula.

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Example  11. We can turn [0, 2π ) into a metric space ([0, 2π ), darc ), using the arc metric  darc : [0, 2π) × [0, 2π ) → R, defined by darc (θ, ϕ) = min{|θ − ϕ|, 2π − |θ − ϕ|}.

Figure 3.

The two metrics on the circle.

2.3. Uniform metric on the space of functions. The next example we can consider is the uniform metric on the space of functions. This is a generalization of the uniform metric d∞ on the finite dimensional spaces n R , so we will use the same notation. Consider the following two sets:

C  = {f  : [0, 1] → R : f  is continuous}, B  = {f  : R → R : f  is bounded}. C  is the set of all continuous functions on the closed interval [0, 1], and B  is the set of all bounded functions on R. (Note that any function f  ∈ C  is also bounded, by the Extreme Value Theorem). Example  12. We can turn C  and B  into metric spaces (called function spaces ) (C , d∞ ) and (B , d∞ ), where we use d∞ to denote the (two distinct) supremum  or uniform metrics  defined as follows: d∞ (f, g ) = sup |f (x) − g (x)|

∀f, g ∈ C ,

x∈[0,1]

d∞ (f, g ) = sup |f (x) − g (x)|

∀f, g ∈ B .

x∈R

Although we abuse notation by referring to both by the same symbol, it should not cause any confusion. Notice that d∞ (f, g ) measures the largest difference between f  and g . We are tempted to define it for all functions R → R, but we run into a problem if  f  or g is unbounded. For example, if  f (x) = x and g (x) = −x, than the formula gives supx∈R |2x| = +∞. Distances, however, have to be real. We will come back to this example in a bit.

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Convergence i n metric spaces

After all the examples we introduced, we will finally generalize some of our definitions about R to metric spaces. This is straightforward, by replacing all mentions of absolute values by distances. Definition 13. Suppose (X, d) is a metric space, xn is a sequence of points  in  X , and  x is another point in  X . We say that  xn converges to x, or that  x is the  limit of  xn , denoted, limd xn = x, if the following statement holds: for  any real  ε > 0, there exists some  N , such that  n > N  implies  d(xn , x) < ε. We say that a sequence  xn in  X  is a  convergent sequence if it has a limit  x ∈ X .

If  d is unambiguous, we can write lim xn as usual, instead of limd xn . Notice that for (R, dstd ) this agrees with usual definition. Exercise  14. For some examples sequences, consider the se 1 n−1 of convergent ( n) n n 2 quence x = (x1 , x2 ) = n , n in R . Show (using Def. 13) that lim x(n) = lim x(n) = lim x(n) = (0, 1). d1

d2

d∞

On the other hand, the sequence θn = 2π[ 2nπ ] on the circle, where [x] denotes the floor of  x, converges for neither d2 nor darc . It keeps going around and around the circle without slowing down. Example  15. Notice that convergence for the metric d∞ in the function spaces C  and B  from Example 12 is equivalent to uniform convergence. This  justifies calling it the uniform metric. On the other hand, pointwise convergence of functions cannot be expressed as convergence in a metric space. (Unless the domain is finite. In that cases the function space is actually Rn ). Somewhat implicit in this discussion is the following result. Theorem 16. If a limit in a metric space exists, it is unique, just as with  R.

Proof. This is a straightforward modification of the usual proof. Suppose ) = z are both limits for xn in (X, d). Let ε = d(y,z y  3 . This is positive by property (M2 ). By the convergences, there are N 1 and N 2 , such that n > N 1 implies d(xn , y) < ε, and n > N 2 implies d(xn , z ) < ε. We can pick a large enough n for which both are satisfied. Then d(y, xn ) + d(xn , z ) = d(xn , y) + d(xn , z ) < 2ε < 3ε = d(y, z ). This contradicts the triangle inequality, i.e. property (M3 ).  Notice that in these example, the existence (and value) of the limit did not depend on the metric. This is not a coincidence, and corresponds to the fact that all our metrics for R2 are equivalent . The two metrics on the circle are also equivalent. Before getting to the notion of equivalent metrics, we need to understand the definition of a limit in two alternative ways. First, we need another definition.

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Definition 17. Let  (X, d) be a metric space. If  x ∈ X  is a point in the  space, and  ε > 0 is arbitrary, we can define the  open ball of radius ε around x, or the  ε–ball around x, to be the set 

Bε (x) = {y ∈ X  : d(x, y) < ε}.

In ( R, dstd ), this is the interval (x−ε, x+ε). In (R2 , d2 ) it is a disk (without its boundary) around x, and in (R3 , d2 ) it is a round three-dimensional ball (without the boundary) around x, of radius ε. Hence the name. For d1 , d2 , d3 , and d∞ on R2 , the balls of radius 1 around (0, 0) are the interiors of the curves in Fig. 1. Theorem 18. Let  xn be a sequence in the metric space  (X, d), and  x ∈ X .

The following three statements are equivalent: (1) limd xn = x, (2) For any  ε > 0, there is an  N , such that  n > N  implies  xn ∈ Bε (x), (3) limn→∞ d(xn , x) = 0 (as a usual limit of real numbers). Proof. The equivalence of (1) and (2) is immediate from the definitions. The equivalence with (3) is a basic consequence of the properties of (usual) convergence for real numbers.  4.

Topological spaces

The statement (2) in Thm. 18 hints at a way of looking at metric spaces in a way less sensitive to the numerical value of distances. The idea is to replace Bε (x) by any set of points “close to x”, or neighborhood of  x. Similar to the notion of distance, we can reduce the notion of a neighborhood to an abstract definition. This is somewhat farther from the intuition, though. Definition 19. A topological space is a pair (X, T  ), where  X  is a set, called  the  space, and  T   is some collection of subsets of  X , each of which we call  an open subset of  X , such that the following conditions are satisfied:

(T1 ) ∅, X  ∈ T  , (T2 ) For  any index set (finite or infinite) I , if  U i ∈ T   for all  i ∈ I , then  i∈I  U i ∈ T  . (T3 ) For any natural number  n, if  U 1 , . . . , Un  ∈ T  , then  U 1 ∩ . . . Un  ∈ T  . A set  V   ⊂ X  is called  closed if its complement  X  \ V   is open. In a more understandable language, (T1 ) says that the empty set, and the space itself are open, (T2 ) says that arbitrary unions of open sets are open, and (T3 ) says that finite  intersections of open sets are open. Note that by basic set algebra, and properties of complements, we see that closed sets satisfy some properties analogous to those for open sets (except unions and intersections are switched). Theorem 20. (T1’ ) ∅ and  X  are closed.

(T2’ ) Any intersection (finite or infinite) of closed sets is closed.

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(T3’ ) Any finite union of closed sets is closed. Example  21. For example, the usual notion of open sets in open sets) makes R into a topological space.

R

(i.e. unions of 

More generally, any metric space can become a topological space. Theorem 22. Let (X, d) be a metric space. Let  T   consist of all sets  U  ⊂ X  with the following property: if  x ∈ U  then there is some real  ε > 0, such that  Bε (x) ⊂ U . Then  (X, T  ) is a topological space. (We call this the  metric topology on  X ).

Proof. We will prove that T   satisfies (T3 ) asthis is the most interesting condition. Suppose U 1 , . . . , Un  ∈ T  , and U  = ni=1 U i . Let x ∈ U . We need to show U  contains a ball around x. Since U  is an intersection, x ∈ U i for all i = 1, . . . , n. But U i ∈ T  , so U i contains some ball Bεi (x) for εi > 0. Let ε = min{ε1 , . . . , εn } > 0. Then Bε (x) ⊂ Bεi (x) ⊂ U i for all i, so Bε (x) ⊂ U . Note that this would fail for an infinite intersection. Instead of a minimum we would need to take an infimum of  εi , but that could be 0 rather than a positive number.  In other words, a set in X  is open if around each of its points it contains an entire open ball. This means that any open set can be written as a (possibly infinite) union of open balls. By property (T2 ), all such unions are open. This can be taken as an alternative definition—a set is open in the metric topology if it is the union of open balls. We can now translate Def. 13 into a statement about open sets without a reference to a metric. This becomes a definition for topological spaces. Definition 23. Let  (X, T  ) be a topological space, xn a sequence in  X , and  x a point in  x. We say that  xn converges to x, or  limT   xn = x if for any  open set  U  ∈ T   which contains  x, there is some  N  such that  n > N  implies  xn ∈ U .

We can easily see that for the metric topology this is equivalent to the previous definition. Theorem 24. Let (X, d) be a metric space, and  (X, T  ) be its metric topology. Then  limd xn = x if and only if  limT   xn = x.

Proof. The backward direction (topological convergence implies metric space convergence) is easy. Just apply Def. 23 to the open set Bε (x). In the other direction, given any open set U  around x, it contains some ball bε (x) ⊂ U  for ε > 0. Now apply Def. 13 to this ε.  Note that while the topological limit agrees with the metric limit for the metric topology, it is not as well behaved for more general topological spaces. For example, the limit is not necessarily unique. Exercise  25. Show that (N, T  ) is a topological space, where N is the natural numbers, and T   consists of all sets U  with finite complements, or is empty.

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That is, U  ∈ T   if it contains all but finitely many of the natural numbers (or none at all). This is called the cofinite topology . Show that limT   n = m for all m ∈ N. I.e. the sequence 1, 2, 3, . . . has all natural numbers as limits under this topology. The next example shows how we can unify finite and infinite limits of  sequences in R into one. Example  26. Consider (R, T  ), where R = R ∪ {−∞, +∞} is the set of extended real numbers, and T   consists of sets U  satisfying the following conditions: (1) If  x ∈ U  and x is real, then there is some ε > 0, such that (x − ε, x + ε) ⊂ U . (2) If  −∞ ∈ U  then there is some M  ∈ R, such that (−∞, M ) ⊂ U . (3) If +∞ ∈ U  then there is some M  ∈ R, such that (M, +∞) ⊂ U . This pair is a topological space, where the notion of convergence is equivalent to our usual three notions of limits. In a sense the intervals [−∞, M ) and (M, +∞] play the roles of open balls around −∞ and +∞, respectively. This is in fact a metric space topology. However, to realize it while having d(−∞, +∞) be finite, we need some not-so-pretty formulas. One option is to set the metric as follows: • For x, y ∈ R, set d(x, y) = | tan−1 x − tan−1 y |. • For x ∈ R, set d(x, −∞) = tan−1 x + π2 . • For x ∈ R, set d(x, +∞) = − tan−1 x + π2 . • Finally, set d(−∞, +∞) = π. The metric topology for d is precisely T  . Exercise  27. Prove the assertions in Example 26. 5.

Closed and compact sets

There are couple of notions that we have encountered before, for sets in R, which can be generalized to metric spaces. One is the notion of closed sets. In fact, we have already given the definition of a closed set. However, we should notice that one of the most important properties of closed sets in R generalizes to this case. Theorem 28. Let  (X, T  ) be a topological space, let  V   ⊂ X  be closed, and  let  xn be a sequence of points in  V  . Suppose  x = limT   xn (recall that  x might  not be unique). Then  x ∈ V  . (In other words, the limit(s) of any sequence  in  V   are also in  V  ).

Proof. The complement X  \ V   is open, i.e. in T  . By the definition of  convergence, there is some N  such that n > N  implies xn ∈ X \ V  . However, xn ∈ V   for all n. This is a contradiction.  Another important notion is that of compact spaces. The general topological definition is somewhat convoluted.

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Definition 29. Suppose  V   is a topological space, or a subset of a topological  space. It is called   compact if the following condition holds: if  V   is contained  in the union  a∈I  U a of some open sets (where the index set  I  could be 

arbitrary), then it is also contained in a finite union of some of these sets: V   ⊂ U a ∪ . . . ∪ U an for some  a1 , . . . , an ∈ I . 1

For metric spaces this is equivalent to something much nicer (and more familiar): Theorem 30. Let  (X, d) be a metric space, and  V   be any subset of  X . Then  V   is compact (with respect to the metric topology) if and only if any  sequence of points in  V   has a subsequence which converges to a point in  V  .

Proof. We will omit the proof.



Note that if  V   is the whole space, then V   is compact if and only if any sequence in V   has a convergent subsequence. Another important theorem (for which we omit the proof) describes all compact sets in Rn (with the metric topology for any d p , p ∈ [1, +∞]). Theorem 31. A set  V   ⊂ Rn is compact if and only if it is closed and  bounded (i.e. there is an  M > 0, such that  d(x, 0) < M  for all  x ∈ V  ).

For example, the circle S 1 with either metric d2 , or darc , is compact. 5.1. Equivalence of metrics. We can get back to the question of why all the metrics d p for p ∈ [1, +∞] give the same notion of convergence on Rn . This is due to the fact that they they are equivalent metrics . Definition 32. Let  d and  d be two functions  X  × X  → R, such that  (X, d) and  (X, d ) are both metric spaces. We say that they are  equivalent metrics

if the following two conditions hold: (1) For all  x ∈ X  and  ε > 0, there is some  δ > 0, such that  d(x, y) < δ  implies  d (x, y ) < ε. (2) For all  x ∈ X  and  ε > 0, there is some  δ > 0, such that  d (x, y) < δ  implies  d(x, y) < ε. Note that the two conditions are symmetric to each other. Roughly, (1) means that points which are close when measured by d are also close when measured by d , and (2) is the reverse statement. More precisely we have the following implications. Theorem 33. If (X, d) and (X, d ) are equivalent metrics, then the following 

statements hold: (1) Any open ball with respect to d around  x contains an open ball  around  x with respect to d, and vice versa. (2) Any open set with respect to d is open with respect to d, and vice  versa. (3) d and  d define the same metric topology on  X .

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Proof. (1) is a restatement of the definition. (2) follows quickly, and (3) is a restatement of (2).  As an immediate corollary we see that limd xn = x if and only if limd xn = 

x.

Exercise  34. Show that the metrics d1 , d2 , and d∞ on R2 are equivalent to each other. Hint: prove that for any real a, b ∈ R, we have 2max{|a|, |b|} ≥ |a| + |b| ≥

 

a2 + b2 ≥ max{|a|, |b|}.

For a bigger challenge, prove that for any fixed n, all the metrics d p for  p ∈ [1, ∞] on Rn are equivalent to each other. Exercise  35. Prove that the metrics d2 and darc on [0, 2π) (from Examples 8 and 11) are equivalent. Hint: use the fact that for any x ∈ [0, π], we have x 2 ≤ sin x ≤ x. Finally, there is another notion of convergence in Rn that seems more familiar from calculus—coordinate-wise convergence. It is also equivalent to metric convergence. (m)

(m)

Theorem 36. Suppose  x(m) = (x1 , . . . , xn ) is a sequence of points in  m n (m) = R , and  x = (x1 , . . . , xn ) is another point in  R . Then  limdp x x (for  (m)

 p ∈ [1, ∞])) if and only if  lim xi

= xi for all  i = 1, . . . , n. In other words, convergence in the metric sense is equivalent to convergence of all individual  coordinates. Proof. In light of Exercise 34, it is enough to prove for one p. We will do it for d∞ . The forward direction is easy. If limd x(m) = x, then d∞ (x(m) , x) con(m) (m) verges to 0. However |xi − xi | ≤ d∞ (x(m) , x) and lim xi = xi by the squeeze theorem, for all i. In the backward direction, suppose the individual coordinates converge. Then for any ε > 0, there are M 1 , . . . , Mn  , such that m > M i implies (m) |xi − xi | < ε. Take M  = max{M 1 , . . . , Mn  }. Then for m > M  all of the inequalities hold, so d∞ (x(m) , x) < ε as well.  ∞

5.2. More topological notions in metric spaces. In addition to convergence and limits of sequences that are a few other notions for metric spaces that are really topological notions. For the definitions, just substitute distance for absolute value. Definition 37. Suppose  (X, dX ) and  (Y, dY  ) are metric spaces, and  f  : X  → Y  is a function. We say that  limx→a f (x) = L (where  a ∈ X  and  L ∈ Y ) if for all  ε > 0 there is some  δ > 0 such that if  dX (x, a) < δ  and  x = a, then  dY  (f (x), L) < ε.

For topological spaces this generalizes to the following.

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Definition 38. Suppose  (X, T  X ) and  (Y, T  Y  ) are topological spaces, and  f  : X  → Y  is a function. We say that limx→a f (x) = L (where  a ∈ X  and  L ∈ Y  ) if for all  U  ∈ T  Y   containing  L, there is some  V   ∈ T  X  containing  a, such that  f (V  ) ⊂ U .

It is easy to see that for metric spaces the two definitions are equivalent, so existence and value of limits are the same for equivalent metrics. Similarly, we can define continuous maps between metric spaces, and generalize to topological spaces. Definition 39. For  (X, dX ) and  (Y, dY  ) metric spaces, a function  f  : X  → Y  is  continuous if for any  x ∈ X  and any  ε > 0, there is some  δ > 0, such  that  dX (x, y) < δ  implies  dY  (f (x), f (y)) < ε. For  (X, T  X ) and  (Y, T  Y  ) topological spaces, a function  f  : X  → Y  is  continuous if for any open set  U  ∈ T  Y  , its preimage is also open, i.e. f −1 (U ) = {x ∈ X  : f (x) ∈ U } ∈ T  X .

Again, for metric spaces the definitions are equivalent, so continuity is the same for equivalent metrics. As an example, it makes sense to talk about continuous maps Rn → Rm without specifying the metric used (any choice of  p ∈ [1, ∞] for Rn , and any (possibly different) choice for Rm give the same result). Thus we can pick and choose the metric for convenience. Recall that in class we mentioned that the Extreme Value Theorem holds for any compact set. In fact it holds for any compact topological space. Theorem 40. Suppose  X  is a compact metric (or topological) space. If  f  : X  → R is continuous (using the standard metric topology on  R), then  the image  f (X ) ⊂ R has a maximum and a minimum.

Proof. For metric spaces, the proof is the same as for a finite closed interval in R. As a sketch, if we assume f (X ) is unbounded, then there is a sequence xn for which f (xn ) diverge to +∞ or −∞. By compactness, there is a convergent subsequence, and by continuity we get a point where f  is infinite. If  f  has a finite supremum, we can pick a sequence of  xn for which f (xn ) approaches the supremum. Again, we can pass to a convergent subsequence to get a point x where f (x) is equal to the supremum. For topological spaces the proof is somewhat different, but we will omit it.  6.

Non-topological aspects of metric spaces

As we saw in the previous section, many of the notions about metric spaces are in fact topological, and not that sensitive to the metric. However, some notions are not topological. This is basically anything which compares distances between different pairs of points, as this cannot be done in an arbitrary topological space. One such notion is that of Cauchy sequences.

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RUMEN ZAREV

Definition 41. A sequence of points  xn in a metric space  (X, d) is called  Cauchy if for all  ε > 0 there is some  N , such that  n , m > N   implies  d(xn , xm ) < ε.

Example  42. A sequence of functions f n in one of the function spaces ( C , dinf ) or (B , dinf ) is Cauchy if and only if it is uniformly Cauchy as defined previously. To really see that being Cauchy is not a topological property, consider the following example. Example  43. Let dexp : R × R → R be the exponential metric  dexp (x, y ) = |ex − ey |. Then (R, dexp ) is a metric space. The metric can be interpreted as follows: as we move toward −∞ points get closer and closer together. In fact the whole interval (−∞, 0) has finite length 1. Recall the standard metric dstd (x, y) = |x − y |. One can check that (R, dstd ) and (R, dexp ) are equivalent metric spaces. Thus any topological notions about them should be the same. Let xn = −n. This sequence is clearly not Cauchy for the standard metric (it is not Cauchy in the usual sense). However, it is  Cauchy for dexp . Indeed, for ε > 0, let N  = − log 2ε . For any n , m > N ,  we get dexp (xn , xm ) = |e−n − e−m | < e−n + e−m < 2e−N  = ε. So xn is Cauchy for one metric, but not for another, equivalent, metric. On the other hand, convergence is  topological. xn does not converge for dstd , so it doesn’t converge for dexp either. This means that we have an example of a sequence in the metric space ( R, dexp ) which is Cauchy but does not converge. We see that unlike in R, in metric spaces Cauchy and convergent are not equivalent. One direction is still true. Any convergent sequence is Cauchy. However the other direction might fail as in the above example. Definition 44. A metric space is called  complete if all Cauchy sequences 

converge. For subspaces of (R, dstd ) (i.e. subsets of  R with the usual notion of  convergence), this completeness is equivalent to completeness in the sense of least upper bounds. Example  45. The two functional spaces B  and C  from Example 12, are complete with respect to d∞ . The completeness of  B  is due to the fact that a uniformly Cauchy sequences is pointwise Cauchy so it pointwise converges. As we have seen it also uniformly converges to the same limit. The limit must be bounded, so it still in B . Similarly, a uniformly Cauchy sequence in C  has a uniform limit. This limit must also be continuous, so it is also in C . The final example of something that can be defined for metric spaces, but not for topological spaces is uniformly continuous functions.

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Definition 46. Suppose  (X, dX ) and  (Y, dY  ) are metric spaces. The function  f  : X  → Y  is  uniformly continuous if for all  ε > 0 there is a  δ > 0 such that  dX (x, y) < δ  implies  dY  (f (x), f (y )) < ε.

As in the usual definition of uniform continuity, δ  depends only on ε, not on x. Thus we can not phrase the condition in term of open sets only, to make it topological.