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Chapter 7 Separation Properties Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and . It is immediately clear from the definitions that XÊXÊXÞ#"! Example 2.2 1) \X is a !BC space iff whenever BÁC, then aa Á that is, different points in \ have different neighborhood systems. 2) If l\l " and \ has the trivial topology, then \ is not a X! space. 3) A pseudometric space Ð\ß .Ñ is metric iff Ð\ß .Ñ is a X! space. 283 Clearly, a metric space is XÐ\ß.ÑXBÁCÞ!!. On the other hand, suppose is and that Then for some %%! either BÂFÐCÑ%% or CÂFÐBÑ. Either way, .ÐBßCÑ , so . is a metric. 4) In any topological space \BµCœwe can define an equivalence relation iffaaBC . and let 1 À \ Ä ] œ \Î µby 1ÐBÑ œ ÒBÓÞGive ] the quotient topology. Then 1 is continuous, onto, open (not automatic for a quotient map!) and the quotient is a X! space: If S\ is open in , we want to show that 1ÒSÓ] is open in , and because ] has the quotient topology this is true iff 1Ò1ÒSÓÓ" is open in \. But 1Ò1ÒSÓÓ " œ ÖB − \ À 1ÐBÑ − 1ÒSÓ× œ ÖB − \ Àfor some C − S, 1ÐBÑ œ 1ÐCÑ× œÖB−\À B is equivalent to some point C in SלS. If ÒBÓ Á ÒCÓ − ], then B is not equivalent to C, so there is an open set S © \ with (say) B − S and C  S. Since 1 is open, 1ÒSÓ is open in ] and ÒBÓ − 1ÒSÓ. Moreover, ÒCÓ Â 1ÒSÓ or else C would be equivalent to some point of S implying C − SÞ ]X is called the X!-identification of \ . This identification turns any space into a ! space by identifying points that have identical neighborhoods. If \X is a ! space to begin with, then 1""1 is and is a homeomorphism. Applied to a XX!! space, the -identification accomplishes nothing. If Ð\ß.Ñis a pseudometric space, the X!-identification is the same as the metric identification discussed in Example VI.5.6Àœ.ÐBßCÑœ!Þ in that caseaaBC iff ww 5) For 3 œ !ß "ß # Àif Ð\ßggg Ñ is a X3 space and ª is a new topology on \, then Ð\ß g Ñ is also a X3 space. Example 2.3 1) (Exercise) It is easy to check that a space \Xis a" space iff for each B−\, ÖB×is closed iff for each , is open and B−\ ÖBל ÖSÀ S B−S× 2) A finite X" space is discrete. 3) Sierpinski space \ œ Ö!ß "× with topology g œ Ögß Ö"×ß Ö!ß "××Ñ is X!" but not X: Ö"× is an open set that contains "! and not ; but there is no open set containing ! and not 1. 4) ‘‘, with the right-ray topology, is XXBC−SœÐBß∞Ñ!" but not : if , then is an open set that contains CB and not ; but there is no open set that contains BC and not . 5) With the cofinite topology, is XX"# but not : in an infinite cofinite space, any two nonempty open sets have nonempty intersection. These separation properties are very well-behaved with respect to subspaces and products. Theorem 2.4 For 3 œ !ß "ß #: 284 a) A subspace of a XX33 space is a space b) If , then is a space iff each is a space. \œα−E \αα Ág \ X33 \ X Proof All of the proofs are easy. We consider only the case 3œ", leaving the other cases as an exercise. w Suppose +Á,−E©\, where \ is a X" space. If Y is an open set in \ containing B but not C, then YœYw ∩E is an open set in E containing B but not C. Similarly we can find an open set Z in E containing CBÞEX but not Therefore is a " space. Suppose is a nonempty space. Each is homeomorphic to a subspace of , so \œα−E \αα X" \ \ each BXαααα is "" (by part a)). Conversely, suppose each \X is and that BÁC−\. Then BÁC for some α. Pick an open set Y\αα in containing B α but not C α. Then YœY α is an open set in \ containing BC but not . Similarly, we find an open set Z\CBin containing but not . Therefore \ is a Xñ" space. Exercise 2.5 Is a continuous image of a XX33 space necessarily a space? How about a quotient? A continuous open image? We now consider a slightly different kind of separation axiom for a space \À formally, the definition is “just like” the definition of X#, but with a closed set replacing one of the points. Definition 2.6 A topological space \JBÂJß is called regular if whenever is a closed set and containing BYZB−YJ©Z, there exist disjoint open sets and such that and . There are some easy variations on the definition of “regular” that are useful to recognize. Theorem 2.7 The following are equivalent for any space \: 285 i) \ is regular ii) if SB is an open set containing , then there exists an open set Y©\ such that B−Y©cl Y©S iii) at each point B−\ there exists a neighborhood base consisting of closed neighborhoods . Proof i)Ê\ ii) Suppose is regular and S is an open set with B−SÞJœ\SLetting , we use regularity to get disjoint open sets YßZ with B − Y and J © Z as illustrated below: Then B−Y©cl Y©S (since cl Y©\Z ). ii)ÊR−B−SœR iii) If aB, then int . By ii), we can find an open set Y so that B−Y©cl Y©S . Since cl Y is a neighborhood of B, the closed neighborhoods of B form a neighborhood base at B. iii)ÊJ i) Suppose is closed and BÂJ. By ii), there is a closed neighborhood OB of such that B−O©\J. We can choose Y œint O and Z œ\O to complete the proof that \ is regular. ñ Example 2.8 Every pseudometric space Ð\ß .Ñ is regular. Suppose +  Jand J is closed. We have a continuous function 0ÐBÑ œ .ÐBß J Ñ for which 0Ð+Ñ œ - ! and 0lJ œ !Þ This gives us disjoint "-- " open sets with +−Y œ0 ÒÐß∞ÑÓ##and J ©Z œ0 ÒÐ∞ß ÑÓ. Therefore \ is regular. At first glance, one might think that regularity is a stronger condition than X#. But this is false: if Ð\ß .Ñ is a pseudometric space but not a metric space, then \ is regular but not even X!. To bring things into line, we make the following definition. Definition 2.9 A topological space \X\ is called a $" space if is regular and X. It is easy to show that XÊX$# ( ÊXÊX "! ): suppose \ is X $ and BÁC−\. Then JœÖC× is closed so, by regularity, there are disjoint open sets YZ, with B−Y and C−ÖCשZ. 286 Caution The terminology varies from book to book. For some authors, the definition of “regular” includes XÀ" for them, “regular” means what we have called “ X$ .” Check the definitions when reading other books. Exercise 2.10 Show that a regular XX!$ space must be (so it would have been equivalent to use “X! ” instead of “XX"$ ” in the definition of “ ”). # Example 2.11 XÊ#$ÎX. We will put a new topology on the set \œ‘ . At each point B−\, let a neighborhood base UB consist of all sets R of the form R œ F% ÐBÑ Ða finite number of straight lines through BÑ ∪ ÖB× for some % !Þ (Check that the conditions in the Neighborhood Base Theorem III.5.2 are satisfied.) With the resulting topology, \FÐBÑ− is called the slotted plane. Note that % UB (because “! ” is a finite number ), so # each FÐBÑ% is among the basic neighborhoods in U‘B so the slotted plane topology on is contains the usual Euclidean topology. It follows that \X is #. The set J œ ÖÐBß !Ñ À B Á !× œ“the B-axis with the origin deleted” is a closed set in \ (why? ).
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