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Chapter 6 Products and Quotients

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Chapter VI Products and Quotients 1. Introduction In Chapter III we defined the product of two topological spaces \] and and developed some simple properties of the product. (See Examples III.5.10-5.12 and Exercise IIIE20. ) Using simple induction, that work generalized easy to finite products. It turns out that looking at infinite products leads to some very nice theorems. For example, infinite products will eventually help us decide which topological spaces are metrizable. 2. Infinite Products and the Product Topology The set \‚] was defined as ÖÐBßCÑÀB−\ßC −]×. How can we define an “infinite product” set ? Informally, we want to say something like \œ Ö\α Àα −E× \œ Ö\αααα Àα −EלÖÐBÑÀB −\ × so that a point BB\ in the product consists of “coordinates αα” chosen from the 's. But what exactly does a symbol like BœÐBÑα mean if there are “uncountably many coordinates?” We can get an idea by first thinking about a countable product. For sets \"# ß \ ß ÞÞÞß \ 8,... we can informally define the product set as a certain set of sequences: ∞ . \œ8œ" \8888 œÖÐBÑÀB −\ × But if we want to be careful about set theory, then a legal definition of \ should have the form \œÖÐBÑ−Y888 ÀB −\ ×. From what “pre-existing set” Y will the sequences in \ be chosen? The answer is easy: we write ∞∞ \œ8œ" \8888 œÖB−Ð 8œ" \Ñ ÀBÐ8ÑœB −\×Þ Thus the elements of ∞ are certain functions (sequences) defined on the index set . This idea 8œ"\8 generalizes naturally to any productÞ Definition 2.1 Let Ö\α Àα − E× be a collection of sets. We define the product set E \œ Ö\αααα Àαα −EלÖB− \ ÀBÐÑ−\ ×. The \'s are called the factors of \. For each th α1α, the function ααÀÖ\À−E×Ä\ α defined by 1 α ÐBÑœB α is called the α -projection map . th For B−\, we write more informally Bαα œBÐα Ñœthe α -coordinate of B and write BœÐB ). Note: the index set EBœÐBÑ might not be ordered. So, even though we use the informal notation α , such phrases as “the first coordinate of BBB,” “the next coordinate in after α,” and “the coordinate in BB preceding αα” may not make sense. The notation ÐBÑis handy but can lead you into errors if you're not careful. 239 A point in is, by definition, a function that “chooses” a coordinate from each set BÖ\À−E× α α Bα in the collection Ö\α Àα − E×. To say that such a “choice function” must exist if all the \α 's are nonempty is precisely the Axiom of Choice. (See the discussion following Theorem I.6.8 .) Theorem 2.2 The Axiom of Choice (AC) is equivalent to the statement that every product of nonempty sets is nonempty. Note: In ZF set theory, certain special products can be shown to be nonempty without using AC. For example, if then ∞ . Without using AC, we can \œ88 ß8œ" \œÖB− ÀBÐ8Ñ−ל precisely describe a point (œ function) in the product for example, the identity function so ∞ . 3œÖÐ7ß8Ñ− ‚ À7œ8× œ8œ" \8 Ág We will often write as . If the indexing set is clearly understood, we may Ö\αα Àα − E×α−E \ E simply write . \α Example 2.3 1) If , then E E œ g Ö\ααα Àαα − E× œ ÖÐα−E \ Ñ À BÐ Ñ − \ × œ Ög×Þ 2) Suppose \œg for some αα −E. Then \œÖÐ \ÑÀBÐÑ−\×ÞE αααα! ! αα−E −E Since BÐα Ñ − \ is impossible, \ œ g. ! αα! αα−E 3) Strictly speaking, we have two different definitions for a finite product\‚\"# : i) \"# ‚\ œÖÐBßBÑÀB "#""## −\ßB −\× (a set of ordered pairs) Ö"ß#× ii) \‚\œÖB−Ð\∪\Ñ"# "# ÀBÐ3Ñ−\× 3 (a set of functions) But there is an obvious way to identify these two sets: the ordered pair ÐB"# ß B Ñ corresponds to the Ö"ß#× function B œ ÖÐ"ß B"# Ñß Ð#ß B Ñ× − Ð\ "# ∪ \ Ñ Þ 4) If , then ∞∞ Eœ Ö\88888 À8− ל8œ" \ œÖB−Ð 8œ" \ Ñ À B −\ × œ ÖÐB"# ß B ß ÞÞÞß B 8 ß ÞÞÞß Ñ À B 8 − \ 8 × œthe set of all sequences ÐB 8 Ñ where B 8 − \ 8 Þ 5) Suppose the 's are identical, say for all . Then \\œ]−EÖ\À−E×αααα α EEE. If , we will sometimes œÖB−Ðα−E \αα Ñ ÀBÐα Ñ−\ לÖB−] ÀB α −]ל] lElœ7 write this product simply as ]œ7 “the product of 7 copies of ]” because the number of factors 7 is often important while the specific index set E is not. Now that we have a definition of the set , we can think about a product topology. We Ö\α Àα − E× begin by recalling the definition and a few basic facts about the “weak topology.” (See Example III.8.6.) 240 Definition 2.4 Let \−EÐ\ßÑ be a set. For each αg, suppose αα is a topological space and that 0À\Ä\αα. The weak topology on \ generated by the collection Y œÖ0Àα α −E× is the smallest topology on \0 that makes all the α's continuous. Certainly, there is at least one topology on \0 that makes all the α's continuous: the discrete topology. Since he intersection of a collection of topologies on \ is a topology (why? ), the weak topology exists we can describe it “from the top down” as { : is a topology on making all the 's \0 gg α continuous×Þ However, this slick description of the weak topology doesn't give a useful description of what sets are open. Usually it is more useful to describe the weak topology on \ “from the bottom up.” to make all the 0α's continuous it is necessary and sufficient that " for each α −E and for each open set Yαα ©\, the set 0α ÒYÓ α must be open. " Therefore the weak topology g is the smallest topology that contains all such sets 0ÒYÓα α and that is " the topology for which ÆαœÖ0α ÒYÓÀααα −EßYopen in \ × is a subbase. (See Example III.8.6. ) Therefore a base for the weak topology consists of all finite intersections of sets from Æ. A typical basic open set has form "] " ... " where each and each is 0ÒY∩0ÒYÓ∩∩0ÒYÓαα"#αα"# α 8 α 8α3 −E Y α 3 open in \α3 . To cut down on symbols, we will use a special notation for these subbasic and basic open sets: " We will write Yαα for 0α ÒY ÓÞ A typical basic open set is then "] " ... " Yœ0ÒY∩0ÒYÓ∩∩0ÒYÓαα"#αα"# α 8 α 8 which we further abbreviate as Y œ Yαα"# ß Y ß ÞÞÞß Y α 8 So B − Y œ Yαα"# ß Y ß ÞÞÞY α 8 iff 0 α 3 ÐBÑ − Y α 3 for each 3 œ "ß ÞÞÞß 8Þ This notation is not standard but it should be because it's very handy. You should verify that to get a base for the weak topology g on \, it is sufficient to use only the sets Yαα"# ß Y ß ÞÞÞß Y α 8 where each Y\is a basic (or even subbasic) open set in . αα3 3 Example 2.5 Suppose Ð\ßg Ñ is any topological space and E©\Þ Let 3ÀEÄ\ be the inclusion map 3Ð+Ñ œ +Þ Then the subspace topology on E is the same as the weak topology generated by Y œÖ3×ÞTo see this, just note that a base for the weak topology is Ö3" ÒYÓÀY open in \× and that 3ÒYÓœY∩E" which is a typical open set in the subspace topology. The following theorem tells us that a map 0\ into a space with the weak topology is continuous iff each composition 0‰0α is continuous. Theorem 2.6 For αg−E, suppose 0ααα À\ÄÐ\ß Ñ and that \ has the weak topology generated by the functions 0^α. Let be a topological space and 0À^Ä\0. Then is continuous iff 0‰0À^Ä\αα is continuous for every α. Proof If 00‰0 is continuous, then each composition α is continuous. Conversely, suppose each " 0‰0α is continuous. To show that 0 is continuous, it is sufficient to show that 0 ÒZÓis open in ^ whenever Z\ZœYY\ is a subbasic open set in (why? ). So let ααwith open in α. Then " " " " 0 ÒZ Ó œ 0 Ò0α ÒYαα ÓÓ œ Ð0 ‰ 0Ñ ÒY α Ó which is open since 0 α ‰ 0 is continuous. ñ 241 Definition 2.7 For each αg−Eßlet Ð\ßαα Ñ be a topological space. The product topology g on the set is the weak topology generated by the collection of projection maps \œÖÀ−E×Þα Y1αα The product topology is sometimes called the “Tychonoff topology.” We always assume that a product space has the product topology unless something else is explicitly stated. \α Since the product topology is a weak topology, a subbase for the product topology consists of all sets " of form Yαα œ1αα ÒYÓ, where −E and Y α is open in \ αÞ A base consists of all finite intersections of such sets À "] ... " Yαααα"8"8 ∩ ÞÞÞ ∩ Y œ11αα"8 ÒY ∩ ∩ ÒY Ó œ ÖB − \ À B − Y for each 3 œ "ß ÞÞÞß 8× αα33 α where for (*) œα−E Yααα Y œ \ααα Á"# ß ß ÞÞÞß α 8 (It is sufficient to use only Y\αα' s which are basic (or even subbasic) open sets in Þ Why?) A basic open set in “depends on only finitely many coordinates” in the following sense: \α B − Yαα"# ß Y ß ÞÞÞß Y α 8 iff B satisfies the finitely many restrictions B α33 − Y α Ð3 œ "ß ÞÞÞß 8Ñ.
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