Chapter II Metric and Pseudometric Spaces 1. Introduction By itself, a set doesn't have any “structure.” For two arbitrary sets EFß and we can ask questions like “Is EœF?” or “Is E equivalent to a subset of F?” but not much more. If we add additional structure to a set, it becomes more interesting. For example, if we define a “multiplication operation” +†, in \ that satisfies certain axioms Ðsuch as +†Ð,†-ÑœÐ+†,ц-ÑÑ, then \ becomes an algebraic structure called a group and a whole area of mathematics known as “group theory” begins. We are not interested in making a set \ into an algebraic system. For topology, we want additional structure on a set \\ for a different purpose: to talk about “nearness” in . This is what we need to discuss topics like “convergence” and “continuity”0 roughly, “ is continuous at +” means that “if B is near +, then 0ÐBÑis near 0Ð+ÑÞ” The simplest way to talk about “nearness” is to equip the set \. with a “distance function” to tell us “how far apart” two elements of \ are. Note: As we proceed we may use ideas taken from elementary analysis, such as the continuity of a function 0À‘‘87 Ä as a source for motivation or examples ß although these ideas will not be carefully defined until later in this chapterÞ 2. Metric and Pseudometric Spaces Definition 2.1 Suppose .À\‚\Ä‘ and that for all BßCßD−\: 1) .ÐBß CÑ ! 2) .ÐBß BÑ œ ! 3) .ÐBß CÑ œ .ÐCß BÑ (symmetry ) 4) .ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ ( the triangle inequality) Such a “distance function” .Ð\ß.Ñ is called a pseudometric on \. The pair is called a pseudometric space. If . also satisfies 5) when BÁC, then .ÐBßCÑ! then .\\ß.Ñ is called a metric on and ( is called a metric space. Of course, every metric space is automatically a pseudometric space. If a pseudometric space Ð\ß .Ñ is not a metric space ßit is because there are at least two points B Á C for which .ÐBß CÑ œ !Þ In most situations this doesn't happen; metrics come up in mathematics more often than pseudometrics. However pseudometrics do occasionally arise in a natural way. Moreover, many definitions and theorems actually only require using properties 1)-4). Therefore we will state our results in terms of pseudometrics rather than metrics in situations where 5) is irrelevant. 61 Example 2.2 1) The usual metric on ‘ is .ÐBß CÑ œ lB ClÞ Clearly, properties 1) - 5) are true. In fact, 1)-5) are chosen so that a metric imitates the usual distance function. 8 2) The usual metric on ‘ is defined as follows: if B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ and 8 are in 8 then # . You should already know that has C œ ÐC"# ß C ß ÞÞÞß C 8 Ñ‘ ß .ÐBß CÑ œ ÐB 3 C 3 Ñ . 3œ" properties 1) - 5). But the details for verifying the triangle inequality are a little tricky, so we will go through the steps. First, we prove another useful inequality. 8 Suppose E œ Ð+"# ß + ß ÞÞÞß + 8 Ñ and F œ Ð, "# ß , ß ÞÞÞß , 8 Ñpoints in ‘ Þ Define 8888 ## ## T ÐAÑ œ Ð+33 A, Ñ œ +33 Ð# + 33 , ÑA Ð , ÑA 3œ" 3œ" 3œ" 3œ" T ÐAÑ is a quadratic function of Aß and T ÐAÑ ! because T ÐAÑ is a sum of squares. Therefore the equation TÐAÑœ ! has at most one real root, so it follows from the quadratic formula that 888 ### Ð# +33 , Ñ %Ð +33 Ñ Ð , Ñ Ÿ !, which gives 3œ" 3œ" 3œ" 888 # "Î# # "Î# |+,lŸÐ33 +Ñ33 Ð , Ñ 3œ" 3œ" 3œ" This last inequality is called the Cauchy-Schwarz inequality. In vector notation it could be written in the form lE † Fl Ÿ llEll † llFllÞ 8 Then if B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ, C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ and D œ ÐD "# ß D ß ÞÞÞß D 8 Ñare in ‘ ß we can calculate 88 ## # .ÐBß DÑ œ ÐB33 D Ñ œ ÐÐB 33 C Ñ ÐC 33 D ÑÑ 3œ" 3œ" 88 8 ## œ ÐB33 CÑ # ÐB 3333 CÑÐC DÑ ÐC 33 DÑ 3œ" 3œ" 3œ" 88 8 ## Ÿ ÐB33 CÑ # lB 3333 CllC Dl ÐC 33 DÑ 3œ" 3œ" 3œ" 888 8 # # "Î# # "Î# # Ÿ ÐBCÑ#Ð33 ÐBCÑÑ 33 Ð ÐBCÑÑ 33 ÐCDÑ 33 3œ" 3œ" 3œ" 3œ" œ Ð.ÐBß CÑ .ÐCß DÑÑ#. Taking the square root of both sides gives .ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ. Example 2.3 We can also put other “unusual” metrics on the set ‘8Þ 62 1) Let . be the usual metric on ‘8w and define . ÐBß CÑ œ "!!.ÐBß CÑ. Then . w is also a metric on ‘‘88wÞÐß.Ñ In , the “usual” distances are stretched by a factor of 100. It is as if we simply changed the units of measurement from meters to centimeters and that change shouldn't matter in any important way. In fact, it's easy to check that if .\ is any metric (or pseudometric) on a set and αα!, then .w œ †. is also a metric (or pseudometric) on \. 8 2) If B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ, C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ are points in ‘ ß define 8 .ÐBßCÑœ>33 lBCl 3œ" 8 It is easy to check that .Ðß.Ñ.>>> satisfies properties 1) - 5) so ‘ is a metric space. We call the taxicab metric on ‘8Þ ( For 8œ#, distances are measured as if you had to move along a rectangular grid of city streets from BC to the taxicab cannot cut diagonally across a city block ). 8 3) If B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ, C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ are points in ‘ ß define ‡ . ÐBß CÑ œmax ÖlB33 C l À 3 œ "ß #ß ÞÞÞß 8× Then Ðß.Ñ‘‘8‡ is also a metric space. We will refer to . ‡as the max metric on 8 Þ ‡ When 8œ"ßof course, .ß.> and . are exactly the same metric on ‘. w‡ 8 We will see later that ..ß.,> are all “equivalent” metrics on ‘ for “topological” purposes. Roughly, this means that whichever of these metrics is used in ‘8ß exactly the same functions are continuous and exactly the same sequences converge. 63 4) The “unit sphere” W"#is the set of points in ‘ that are at distance 1 from the origin. Sketch #‡w the unit sphere in ‘ using the metrics .ß .> ß ., and . œ "!!.Þ Since there are only two coordinates, we will write a point in ‘# in the usual way as ÐBß CÑ rather than ÐB"# ß B Ñ. For ..ß, we get For > we get "" W œ ÖÐBß CÑ À .Ð ÐBß CÑß Ð!ß !ÑÑ œ "× W œ ÖÐBß CÑ À .> Ð ÐBß CÑß Ð!ß !ÑÑ œ "× œÖÐBßCÑÀB## C œ"× œÖÐBßCÑÀlBllClœ"× For .‡, we get W"‡ œ ÖÐBß CÑ À . Ð ÐBß CÑß Ð!ß !ÑÑ œ "× œ ÖÐBß CÑ Àmax ÖlBlß lCl× œ "× Of course for the metric .w" œ "!!., W has the same shape as for the metric ., but the sphere is reduced in size by a scaling factor of "!!. w‡ 8 Switching among the metrics .ß ., .> ß . produces unit spheres in ‘ with different sizes and shapes. In other words, changing the metric on ‘8 may cause dramatic changes in the geometry of the space for example, “areas” may change and “spheres” may no longer be “round.” Changing the metric can also affect smoothness features of the space ÐÑspheres may turn out to have sharp corners . But it turns w‡ out, as mentioned earlier, that .ß ., .> and . are “equivalent” for “topological purposes.” For topology, “size,” “geometrical shape,” and “smoothness” don't matter. 64 When talking about ‘‘8ß.the usual metric is the default that is, we always assume that 8, or any subset of ‘8, has the usual metric . unless a different metric is explicitly stated. Example 2.4 In each part, you should verify that . satisfies the properties of a pseudometric or metric. 1) For a set \, define .ÐBßCÑœ!for all BßC−\Þ We call . the trivial pseudometric on \: all distances are 0. (Under what circumstances is this . a metric? ) !BœCif 2) For a set \.ÐBßCÑœ, define . We call . the discrete unit metric on \. "BÁCif To verify the triangle inequality: for points Bß Cß D − \, .ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ certainly is true if BœD; and if BÁD, then .ÐBßDÑœ" and .ÐBßCÑ.ÐCßDÑ ". Definition 2.5 Suppose Ð\ß .Ñ is a pseudometric space, that B! − \ and % !. Then FÐBÑœÖB−\À.ÐBßBÑ% !!% × is called the ball of radius % with center at B !. If there exists an % ! such that FÐBÑœÖB×% !!, then we say that B ! is an isolated point in (\ß.ÑÞ Example 2.6 8 1) In ‘%%‘, FÐBÑœÐB%%!!! ßB Ñ. More generally, FÐBÑ ! in is just the usual spherical 8 ball with radius % and center at BÞ.ß!> (not including the boundary surface) If the metric is used in ‘ thenFÐBÑ% ! is the interior of a “diamond-shaped” region centered at B!. ÐSee the earlier sketches of "# " W: in Б , .>" Ñ , F ÐÐ!ß !ÑÑ is the region “inside” the diamond-shaped W ÞÑ " In \ œ Ò!ß "Ó with the usual metric . , then F" Ð!Ñ œ Ò!ß Ñ, F"# Ð!Ñ œ Ò!ß "Ñ ß F Ð!Ñ œ Ò!ß "ÓÞ # # 2) If .is the trivial pseudometric on \and B!! −\ßthen FÐBÑœ\% for every % !Þ ÖB! ×if % Ÿ " $ÑIf .