Chapter VII Separation Properties 1. Introduction “Separation” refers here to whether objects such as points or disjoint closed sets can be enclosed in disjoint open sets. In spite of the similarity of terminology, “separation properties” have no direct connection to the idea of “separated sets” that appeared in Chapter 5 in the context of connected spaces. 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 hypotheses for “more separation” are added, spaces 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 original literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , ,and X % , each one stronger than its predecessor in the list. Once these had become common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: X" Þ 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 YB−YCÂY with , or there exists an open set Zwith C−ZBÂZ, 2) X"-space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set Zwith BÂYßC−Z 3) X#-space (or, Hausdorff space) if, whenever BÁC−\, there exist disjoint open sets Y and Z in \ such that B−Y and C−Z . It is immediately clear from the definitions that X# ÊX " ÊXÞ ! Example 2.2 1) \X is a !-space if and only if: whenever BÁC, then aB Á a C that is, different points in \ have different neighborhood systems. 283 2) If \ has the trivial topology and l\l"\, then is not a X !-space. 3) A pseudometric space Ð\ß.Ñ is a metric space in and only if Ð\ß.Ñ is a X !-space. Clearly, a metric space is X!. On the other hand, suppose Ð\ß.ÑX is ! and that BÁCÞ Then for some % ! either BÂFÐCÑ% or CÂFÐBÑ % . Either way, .ÐBßCÑ % , so . is a metric. 4) In any topological space \ we candefine an equivalence relation BµCiffaB œ a C . Let 1À\Ä\εœ] by 1ÐBÑœÒBÓÞGive ] 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 1" Ò1ÒSÓÓ is open in \. But 1 " Ò1ÒSÓÓ œÖB−\À1ÐBÑ−1ÒSÓלÖB−\Àfor some C−S, 1ÐBÑœ1ÐCÑ× œÖB−\ÀB is equivalent to some point CSלS in . If ÒBÓÁÒCÓ−], then B is not equivalent to C, so there is an open set S©\ with (say) B−S and CÂS. Since 1 is open, 1ÒSÓ is open in ] and ÒBÓ−1ÒSÓ . Moreover, ÒCÓÂ1ÒSÓ or else C would be equivalent to some point of Simplying C−SÞ ] is called the X!-identification of \ . This identification turns any space into a X ! -spaceby identifying points that have identical neighborhoods. If \ is a X!-space to begin with, then 1 is one- to-one and 1 is a homeomorphism. Applied to a X! space, the X ! -identification accomplishes nothing. If Ð\ß.Ñis a pseudometric space, the X !-identification is the same as the metric identification discussed in Example VI.5.6 because, in that case,aBœ a C if and only if .ÐBßCÑœ!Þ w w 5) For 3œ!ß"ß#ÀÐ\ßÑif g is a X3 space and gg ª is a new topology on \, then Ð\ßÑ g is also a X3 space. Example 2.3 1) (Exercise) It is easy to check that a space \is a X " space iff for each B − \, ÖB× is closed iff for each B−\ÖBל, + ÖSÀSis open and B−S× 2) A finite X" space is discrete. 3) Sierpinski space \œÖ!ß"× with topology g œÖgßÖ"×ßÖ!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 X! but not X " : if B+C−‘ , then SœÐBß∞Ñ 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 X" but not X # because, in an infinite cofinite space, any two nonempty open sets have nonempty intersection. 284 These separation properties are very well-behaved with respect to subspaces and products. Theorem 2.4 For 3 œ !ß"ß# : a) A subspace of a X3-space is a X 3 -space b) If \œ#α−E \Ágα , then \X is a 3-space iff each \α is a X 3 -space. Proof All of the proofs are easy. We consider here only the case 3 œ " , leaving the other cases as an exercise. w a) Suppose +Á,−E©\, where \X is a " space. If Y is an open set in \ containing B but not C , then YœY∩Ew is an open set in E containing BC but not . Similarly we can find an open set ZE in containing C but not BÞ Therefore EXis a "-space. b) Suppose \œ\#α−E α is a nonempty X"-space. Each \α is homeomorphic to a subspace of \ , so, by part a), each \XÞα is " Conversely, suppose each \Xα is " and that BÁC−\. Then BÁCα α for some α. Pick an open set Y\αα in containing B αα but not C. Then Yœ+Y α is an open set in \BCcontaining but not . Similarly, we find an open set Z\CBin containing but not . Therefore \ is a X"-space. ñ Exercise 2.5 Is a continuous image of a X3-space necessarily a X 3 -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 \ is called regular if whenever J is a closed set and BÂJß there exist disjoint open sets Y and Z such that B−Y and J©Z . 285 There are some easy equivalents of the definition of “regular” that are useful to recognize. Theorem 2.7 The following are equivalent for any space \: i) \ is regular ii) if S is an open set containing B, 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−SÞLetting Jœ\S , 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− iii) If aB, then B−SœRint . By ii), we can find an open set Y so that B−Y©Y©S©RÞcl Since cl Y is a neighborhood of B, the closed neighborhoods of B form a neighborhood base at B. iii)Ê i) Suppose J is closed and BÂJ. By ii), there is a closed neighborhood OB of such that B−O©\J. We can choose YœOint and Zœ\O to complete the proof that \ is regular. ñ Example 2.8 Every pseudometric space Ð\ß.Ñ is regular. Suppose +ÂJand J is closed. We have a continuous function 0ÐBÑœ.ÐBßJÑ for which 0Ð+Ñœ-! and 0lJœ!Þ This gives us disjoint "- " - open sets with +−Yœ0 ÒÐß∞ÑÓ# and J©Zœ0 ÒÐ∞ßÑÓ# . 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 !. 286 To bring things into line, we make the following definition. Definition A topological space \ is called a X\$-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 . Caution 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 X! space must be X $ ( so it would have been equivalent to use “X! ” instead of “ X " ” in the definition of “ X $ ” ). # Example 2.11 XÊX#Î $ . We will put a new topology on the set \œ‘ . At each point :−\ , let a neighborhood base U: consist of all sets R of the form RœFÐ:ÑÐ% a finite number of straight lines through :Ñ∪Ö:× for some % !Þ (Check that the conditions in the Neighborhood Base Theorem III.5.2 are satisfied. ) With the resulting topology, \ is called the slotted plane . Note that F% Ð:Ñ − U: ( because “! ” is a finite # number ), so each F% Ð:Ñ is among the basic neighborhoods in U: so the slotted plane topology on ‘ contains the usual Euclidean topology. It follows that \ is X #. The set JœÖÐBß!ÑÀBÁ!ל“the B-axis with the origin deleted” is a closed set in \ (why? ).