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Chapter 5 Connectedness

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Chapter 5 Connectedness Chapter V Connected Spaces 1. Introduction In this chapter we introduce the idea of connectedness. Connectedness is a topological property quite different from any property we considered in Chapters 1-4. A connected space \ need not have any of the other topological properties we have discussed so far. Conversely, the only topological properties that imply “\l\lŸ\ is connected” are very extreme such as “ 1” or “ has the trivial topology.” 2. Connectedness Intuitively, a space is connected if it is all in one piece; equivalently a space is disconnected if it can be written as the union of two nonempty “separated” pieces. To make this precise, we need to decide what “separated” should mean. For example, we think of ‘‘ as connected even though can be written as the union of two disjoint pieces: for example, ‘ œE∪Fß where E œ Ð ∞ß !Ó and F œ Ð!ß ∞Ñ. Evidently, “separated” should mean something more than “disjoint.” On the other hand, if we remove the point ! to “cut” ‘, then we probably think of the remaining space \ œ‘ Ö!× as “disconnected.” Here, we can write \ œ E ∪ F, where E œ( ∞ß !Ñ and FœÐ!ß∞Ñ. E and F are disjoint, nonempty sets and (unlike EF and in the preceding paragraph) they satisfy the following (equivalent) conditions: i) EFand are open in \ ii) EFand are closed in \ iii) ÐF ∩cl\\ EÑ ∪ ÐE ∩ cl FÑ œ g that is, each of E and F is disjoint from the closure of the other. (This is true, in fact, even if we use cl‘ instead of cl\ .) Condition iii) is important enough to deserve a name. Definition 2.1 Suppose EF and are subspaces of Ð\ßÑEFg . and are called separated if each is disjoint from the closure of the otherÐF∩EÑ∪ÐE∩FÑœg that is, if cl\\ cl . It follows immediately from the definition that i) separated sets must be disjoint, and ii) subsets of separated sets are separated: if Eß F are separated, G © E and H©F, then Gand H are also separated. Example 2.2 1) In‘ , the sets E œ Ð ∞ß !Ó and F œ Ð!ß ∞Ñ are disjoint but not separated. Likewise in ‘#####, the sets EœÖÐBßCÑÀB C Ÿ"×and FœÖÐBßCÑÀÐB 2 Ñ C "× are disjoint but not separatedÞ 213 2) The intervals E œ Ð ∞ß !Ñ and F œ Ð!ß ∞Ñare separated in ‘ but cl‘‘ E ∩ cl F Á gÞ The same is true for the open balls EœÖÐBßCÑÀB## C "×and F œ ÖÐBß CÑ À ÐB 2 Ñ## C "× in ‘ #. The condition that two sets are separated is stronger than saying they are disjoint, but weaker than saying that the sets have disjoint closures. Theorem 2.3 In any space Ð\ßg Ñ, the following statements are equivalent: 1) g\ and are the only clopen sets in \ 2) if E©\ and Fr Eœg , then Eœg or Eœ\ 3) \ is not the union of two disjoint nonempty open sets 4) \ is not the union of two disjoint nonempty closed sets 5) \ is not the union of two nonempty separated sets. Note: Condition 2) is not frequently used. However it is fairly expressive: to say that FrEœg says that no point B in \ can be “approximated arbitrarily closely” from both inside and outside Eso, in that sense, Eand Fœ\E are pieces of \ that are “separated” from each other. Proof 1)ÍEEœg 2) This follows because is clopen iff Fr (see Theorem II.4.5.3 ). 1)Ê\œE∪FEF 3) Suppose 3) is false and that where , are disjoint, nonempty and open. Since \EœF is open and nonempty, we have that E is a nonempty proper clopen set in \, which shows that 1) is false. 3)Í 4) This is clear. 4)Ê\œE∪FEßF 5) If 5) is false, then , where are nonempty and separated. Since clF∩Eœgß we conclude that cl F©F , so F is closed. Similarly, E must be closed. Therefore 4) is false. 5)ÊE 1) Suppose 1) is false and that is a nonempty proper clopen subset of \. Then Fœ\E is nonempty and clopen, so E and F are separated. Since \œE∪F, 5) is false. ñ Definition 2.4 A space Ð\ßg Ñ is connected if any (therefore all) of the conditions 1) - 5) in Theorem 2.3 hold. If G©\, we say that G is connected if G is connected in the subspace topology. According to the definition, a subspace G©\ is disconnected if we can write GœE∪F, where the following (equivalent) statements are true: 1) EFand are disjoint, nonempty and open in G 2) EFand are disjoint, nonempty and closed in G 3) EF and are nonempty and separated in G. If GEßFG is disconnected, such a pair of sets will be called a disconnection or separation of . The following technical theorem and its corollary are very useful in working with connectedness in subspaces. 214 Theorem 2.5 Suppose EßF©G©\ÞEFThen and are separated in G iff EF and are separated in \. Proof clG\FœG∩ cl F (see Theorem III.7.6 ), so E∩cl G Fœg iff E∩Ðcl \ F∩GÑœg iffÐE∩GÑ∩ cl\\ Fœg iff E∩cl FœgÞ Similarly, F∩cl G\ Eœg iff F∩cl EœgÞ ñ Caution: According to Theorem 2.5, GGœE∪FEF is disconnected iff where and are nonempty separated set in GGœE∪FEF iff where and are nonempty separated set in \. Theorem 2.5 is very useful because it means that we don't have to distinguish here between “separated in G\” and “separated in ” because these are equivalent. In contrast, when we say that GG is disconnected if is the union of two disjoint, nonempty open (or closed ) sets EßF in G, then phrase “in GEF\”cannot be omitted: the sets , might not be open (or closed ) in . "" " For example, suppose \ œ Ò!ß "Ó and G œ Ò!ß## Ñ ∪ Ð ß "Ó . The sets E œ Ò!ß # Ñ and " FœÐ# ß"Ó are open, closed and separated in G. By Theorem 2.5, they are also separated in ‘ but they are neither open nor closed in ‘. Example 2.6 1) Clearly, connectedness is a topological property. More generally, suppose 0À\Ä] is continuous and onto. If F]0ÒFÓ is proper nonempty clopen set in , then " is a proper nonempty clopen set in \. Therefore a continuous image of a connected space is connected. 2) A discrete space \l\lŸ" is connected iff . In particular, ™ and are not connected. 3) is not connected since we can write as the union of two nonempty separated sets: œÖ;− À;## #×∪Ö;− À; #×. Similarly, we can show is not connected. More generally suppose G©‘ and that G is not an interval. Then there are points +D, where +ß,−G but DÂGÞThen ÖB−GÀBDלÖB−GÀBŸD× is a nonempty proper clopen set in GG. Therefore is not connected. In fact, a subset GG of ‘ is connected iff is an interval. It is not very hard, using the least upper bound property of ‘‘, to prove that every interval in is connected. (Try it as an exercise! ) We will give a short proof soon (Corollary 2.12) using a different argument. 4) (The Intermediate Value Theorem ) If \0À\Ä is connected and ‘ is continuous, then ranÐ0Ñ is connected (by part 1 ) so ran Ð0Ñ is an interval ( by part 3 ). Therefore if +ß , − \ and 0Ð+ÑD0Ð,Ñ, there must be a point -−\ for which 0Ð-ÑœDÞ 5) The Cantor set G is not connected (since it is not an interval). But much more is true. Suppose Bß C − E © G and that B CÞSince G is nowhere dense (see IV.10 ), the interval ÐBß CÑ ©Î G, so we can choose DÂG with BDC. Then FœÐ∞ßDÑ∩E œÐ∞ßDÓ∩E is clopen in E, and F contains B but not Cso E is not connected. It follows that every connected subset of G contains at most one point. A space Ð\ßg Ñ is called totally disconnected every connected subset E satisfies lEl Ÿ "Þ The spaces ™ ß ß and are other examples of totally disconnected spaces. 6) \ is connected iff every continuous 0 À \ Ä Ö!ß "× is constant: certainly, if 0 is continuous and not constant, then 0" ÒÖ!×Ó is a proper nonempty clopen set in \ so \ is not 215 connected. Conversely, if \Eis not connected and is a proper nonempty clopen set, then the characteristic function ;E À \ Ä Ö!ß "× is continuous but not constant. Theorem 2.7 Suppose 0 À \ Ä ]. Let > œ ÖÐBß CÑ − \ ‚ ] À C œ 0ÐBÑ× œ“the graph of 0Þ” If 00 is continuous, then graph of is homeomorphic to the domain of 0; in particular, the graph of a continuous function is connected iff its domain is connected. Proof We want to show that \2À\Ä is homeomorphic to >>. Let be defined by 2ÐBÑ œ ÐBß 0ÐBÑÑÞ Clearly 2 is a one-to-one map from \ onto >. Let + − \ and suppose ÐY ‚ Z Ñ ∩> is a basic open set containing 2Ð+Ñ œ Ð+ß 0Ð+ÑÑÞ Since 00Ð+Ñ−ZßS\+ is continuous and there exists an open set in containing and such that 0ÒSÓ © Z Þ Then + − Y ∩ S, and 2ÒY ∩ SÓ © ÐY ‚ Z Ñ ∩>, so 2 is continuous at +. If Y\2ÒYÓœÐY‚]Ñ∩2 is open in , then >> is open in , so is open. Therefore 2 is a homeomorphism. ñ Note: It is not true that a function 0 with a connected graph must be continuous. See Example 2.22. The following lemma makes a simple but very useful observation.
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