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 “\ is connected” are very extreme  such as “l\l Ÿ 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, ‘ œ E ∪ 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 F œ Ð!ß ∞Ñ. E and F are disjoint, nonempty sets and (unlike E and F in the preceding paragraph) they satisfy the following (equivalent) conditions: i) E and F are open in \ ii) E and F 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 E and F are subspaces of Ð\ß g Ñ. E and F are called separated if each is disjoint from the closure of the other  that is, if ÐF ∩ cl\ EÑ ∪ ÐE ∩ cl\ FÑ œ g. 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 G and 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 E œ ÖÐBß CÑ À B#  C# Ÿ "× and F œ ÖÐBß CÑ À ÐB  2Ñ#  C#  "× are disjoint but not separatedÞ

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2) The intervals E œ Ð  ∞ß !Ñ and F œ Ð!ß ∞Ñ are separated in ‘ but cl‘ E ∩ cl‘ F Á gÞ The same is true for the open balls E œ ÖÐBß CÑ À B#  C#  "× and F œ ÖÐBß CÑ À ÐB  2Ñ#  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\ß g Ñ, the following statements are equivalent: 1) 2) 3) 4) 5)

g and \ are the only clopen sets in \ if E © \ and Fr E œ g, then E œ g or E œ \ \ is not the union of two disjoint nonempty open sets \ is not the union of two disjoint nonempty closed sets \ is not the union of two nonempty separated sets.

Note: Condition 2) is not frequently used. However it is fairly expressive: to say that Fr E œ g says that no point B in \ can be “approximated arbitrarily closely” from both inside and outside E  so, in that sense, E and F œ \  E are pieces of \ that are “separated” from each other. Proof 1) Í 2) This follows because E is clopen iff Fr E œ g (see Theorem II.4.5.3). 1) Ê 3) Suppose 3) is false and that \ œ E ∪ F where E, F are disjoint, nonempty and open. Since \  E œ F 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 5) If 5) is false, then \ œ E ∪ F , where Eß F are nonempty and separated. Since cl F ∩ E œ gß we conclude that cl F © F , so F is closed. Similarly, E must be closed. Therefore 4) is false. 5) Ê 1) Suppose 1) is false and that E is a nonempty proper clopen subset of \ . Then F œ \  E is nonempty and clopen, so E and F are separated. Since \ œ E ∪ F , 5) is false. ñ Definition 2.4 A space Ð\ß g 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 G œ E ∪ F , where the following (equivalent) statements are true: 1) E and F are disjoint, nonempty and open in G 2) E and F are disjoint, nonempty and closed in G 3) E and F are nonempty and separated in G . If G is disconnected, such a pair of sets Eß F will be called a disconnection or separation of G .

The following technical theorem and its corollary are very useful in working with connectedness in subspaces.

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Theorem 2.5 Suppose Eß F © G © \Þ Then E and F are separated in G iff E and F are separated in \ . Proof clG F œ G ∩ cl\ F (see Theorem III.7.6), so E ∩ clG F œ g iff E ∩ Ðcl \ F ∩ GÑ œ g iff ÐE ∩ GÑ ∩ cl \ F œ g iff E ∩ cl \ F œ gÞ Similarly, F ∩ clG E œ g iff F ∩ cl\ E œ gÞ ñ Caution: According to Theorem 2.5, G is disconnected iff G œ E ∪ F where E and F are nonempty separated set in G iff G œ E ∪ F where E and F 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 G is disconnected if G is the union of two disjoint, nonempty open (or closed) sets Eß F in G, then phrase “in G ” cannot be omitted: the sets E, F 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 " F œ Ð # 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 0 À \ Ä ] is continuous and onto. If F is proper nonempty clopen set in ] , then 0 " ÒFÓ is a proper nonempty clopen set in \ . Therefore a continuous image of a connected space is connected. 2) A discrete space \ is connected iff l\l Ÿ ". In particular,  and ™ are not connected. 3)  is not connected since we can write  as the union of two nonempty separated sets:  œ Ö; −  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 D  GÞ Then ÖB − G À B  D× œ ÖB − G À B Ÿ D× is a nonempty proper clopen set in G . Therefore Gis not connected. In fact, a subset G of ‘ is connected iff G 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 \ is connected and 0 À \ Ä ‘ is continuous, then ran Ð0 Ñ is connected (by part 1) so ranÐ0 Ñ is an interval (by part 3). Therefore if +ß , − \ and 0 Ð+Ñ  D  0 Ð,Ñ, there must be a point - − \ for which 0 Ð-Ñ œ DÞ 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  CÞ Since G is nowhere dense (see IV.10), the interval ÐBß CÑ © Î G , so we can choose D  G with B  D  C. Then F œ Ð  ∞ß DÑ ∩ E œ Ð  ∞ß DÓ ∩ E is clopen in E, and F contains B but not C  so E is not connected. It follows that every connected subset of G contains at most one point. A space Ð\ß g 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

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connected. Conversely, if \ is not connected and E 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 0 ÐBÑ× œ “the graph of 0 Þ” If 0 is continuous, then graph of 0 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 \ is homeomorphic to >. Let 2 À \ Ä > be defined by 2ÐBÑ œ ÐBß 0 ÐBÑÑÞ 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 0 Ð+ÑÑÞ Since 0 is continuous and 0 Ð+Ñ − Z ß there exists an open set S in \ containing + and such that 0 ÒSÓ © Z Þ Then + − Y ∩ S, and 2ÒY ∩ SÓ © ÐY ‚ Z Ñ ∩ >, so 2 is continuous at +. If Y is open in \ , then 2ÒY Ó œ ÐY ‚ ] Ñ ∩ > is open in >, so 2 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. Lemma 2.8 Suppose Q ß R are separated subsets of \ . If G © Q ∪ R and G is connected, then G © Q or G © R . Proof G ∩ Q and G ∩ R are separated (since G ∩ Q © Q and G ∩ R © R ) and G œ ÐG ∩ Q Ñ ∪ ÐG ∩ R Ñ. But G is connected so ÐG ∩ Q Ñ and ÐG ∩ R Ñ cannot form a disconnection of G . Therefore either G ∩ Q œ g (so G © R Ñ or G ∩ R œ g (so G © Q ). ñ

The next theorem and its corollaries are simple but powerful tools for proving that certain sets are connected. Roughly, the theorem states that if we have one “central ” connected set G and other connected sets none of which is separated from G , then the union of all the sets is connected. Theorem 2.9 Suppose G and Gα (α − M ) are connected subsets of \ and that for each α, Gα and G are not separated. Then W œ G ∪ Gα is connected. Proof Suppose that W œ Q ∪ R where Q and R are separated. By Lemma 2.8, either G © Q or G © R . Without loss of generality, assume G © Q . By the same reasoning we conclude that for each α, either Gα © Q or Gα © R Þ But if some Gα © R , then G and Gα would be separated. Hence every Gα © Q . Therefore R œ g and the pair Q ß R is not a disconnection of W. ñ

Corollary 2.10 Suppose that for each α − M , Cα is a connected subset of \ and that for all α Á " − M , Cα ∩ G" Á g. Then ÖCα À α − M× is connected.

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Proof If M œ gß then ÖCα À α − M× œ g is connected. If M Á g, pick an α! − M and let Gα! be the “central set” G in Theorem 2.9. For all α − M , Gα ∩ Gα! Á g, so Gα and Gα! are not separated. By Theorem 2.9, ÖCα À α − M× is connected. ñ Corollary 2.11 For each 8 − , suppose G8 is a connected subset of \ and that G8 ∩ Gn" Á g. Then ∞ 8œ" G8 is connected.

Proof Let E8 œ 85œ" G5 . Corollary 2.10 (and simple induction) shows that the E8 's are connected. Then g Á E" © E# © ÞÞÞ © E8 © ÞÞÞ Another application of Corollary 2.10 gives ∞ that ∞ 8œ" E8 œ 8œ" G8 is connected. ñ Corollary 2.12 Let M © ‘Þ Then M is connected iff M is an interval. In particular, ‘ is connected, so ‘ and g are the only clopen sets in ‘. Proof We have already shown that if M is not an interval, then M is not connected (Example 2.6.3). So suppose M is an interval. If M œ g or M œ Ö is homeomorphic to Ð!ß "Ó so > is connected.

> is sometimes called the “topologist's sine curve.” Because cl > œ > ∪ ÐÖ!× ‚ Ò  "ß "ÓÑ, Corollary 2.20 gives that > ∪ E is connected for any set E © Ö!× ‚ Ò  "ß "ÓÞ In particular, >0 œ > ∪ ÖÐ!ß !Ñ× Ðthe graph of 0 Ñ is connected. Therefore, a function 0 with a connected graph need not be continuous. However, it is true that if the graph of a function 0 À ‘ Ä ‘ is a closed connected subset of ‘# , then 0 is continuous. (The proof is easy enough to read: see C.E. Burgess, Continuous Functions and Connected Graphs, The American Mathematical Monthly, April 1990, 337-339.)

3. Path Connectedness and Local Path Connectedness

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In some spaces \ , every pair of points can be joined by a path in \ . This seems like a very intuitive way to describe “connectedness”. However, this property is actually stronger than our definition for a connected space. . Definition 3.1 A path in \ is a continuous map 0 À Ò!ß "Ó Ä \Þ The path starts at its initial point 0 Ð!Ñ and ends at its terminal point 0 Ð"ÑÞ We say 0 is a path from 0 Ð!Ñ to 0 Ð"ÑÞ

Sometimes it helps to visualize a path by thinking of a point moving in \ from 0 Ð!Ñ to 0 Ð"Ñ with 0 Ð>Ñ representing its position at “time” > − Ò!ß "Ó. Remember, however, that the path, by definition, is the function 0 , not the set ranÐ0 Ñ © \ . To illustrate the distinction: suppose 0 is a path from B to C. Then the function 1 À Ò!ß "Ó Ä \ defined by 1Ð>Ñ œ 0 Ð"  >Ñ is a different path (running in the “opposite direction,” from C to B), even though ranÐ0 Ñ œ ranÐ1ÑÞ

Definition 3.2 A topological space \ is called path connected if, for every pair of points Bß C − \, there is a path from B to C in \ . Note: \ is called arcwise connected if, for every pair of points Bß C − \ , there exists a homeomorphism 0 À Ò!ß "Ó Ä \ with 0 Ð!Ñ œ B and 0 Ð"Ñ œ C. Such a path 0 is called an arc from B to C. If a path 0 in a Hausdorff space \ is not an arc, the reason must be that 0 is not one-to-one (why?). It can be proven that a Hausdorff space is path connected iff \ is arcwise connected. Therefore some books use “arcwise connected” to mean the same thing as “path connected.” Theorem 3.3 A path connected space \ is connected. Proof g is connected, so assume \ Á g and choose a point + − \ . For each B − \ , there is a path 0B from + to B. Let GB œ ranÐ0B Ñ. Each GB is connected and contains +. By Corollary 2.10, \ œ ÖGB À B − \× is connected. ñ

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Sometimes path connectedness and connectedness are equivalent. For example, a subset M © ‘ is connected iff M is an interval iff M is path connected. But in general, the converse to Theorem 3.,3 is false as the next example shows. Consider 0 ÐBÑ œ 

sin 1B !  B Ÿ " . In Example 2.22, we showed that the ! Bœ! graph >0 is connected. However, we claim that there is no path in >0 from Ð!ß !Ñ to Ð"ß !Ñ and therefore >0 is not path connected. Example 3.4

Suppose, on the contrary, that 2 À Ò!ß "Ó Ä >0 is a path from Ð!ß !Ñ to Ð"ß !ÑÞ For > − Ò!ß "Ó, write 2Ð>Ñ œ Ð2" Ð>Ñß 2# Ð>ÑÑ − >0 . 2" and 2# are continuous (why? ). Since Ò!ß "Ó is compact, 2 is uniformly continuous (Theorem IV.9.6) so we can choose $"  ! for which l?  @l  $" Ê .Ð2Ð?Ñß 2Ð@ÑÑ  " Ê l2# Ð?Ñ  2# Ð@Ñl  "Þ We have ! − 2" ÐÐ!ß !ÑÑ. Let >‡ œ sup 2" Ð!ß !ÑÞ Then ! Ÿ >‡  ". Since 2" Ð!ß !Ñ is a closed set, >‡ − 2" Ð!ß !Ñ so 2Ð>‡ Ñ œ Ð!ß !ÑÞ (We can think of t‡ as the last “time” that the path 2 goes through the origin). Choose a positive $  $" so that ! Ÿ >‡  >‡  $  ". Since 2" Ð>‡ Ñ œ ! and 2" Ð>‡  $ Ñ  !, we can choose a positive integer R for which ! œ 2" Ð>‡ Ñ Ÿ

# R "



# R

 2" Ð>‡  $ ÑÞ

By the Intermediate Value Theorem, there exist points ?ß @ − Ð>‡ ß >‡  $ Ñ where 2" Ð?Ñ œ

# R "

and 2" Ð@Ñ œ R# . Then 2# Ð?Ñ œ sin ÐR # "Ñ1 and 2# Ð@Ñ œ sin R#1 , so l2# Ð?Ñ  2# Ð@Ñl œ ". But this is impossible since l?  @l  $  $" and therefore l2# Ð?Ñ  2# Ð@Ñl  "Þ ñ Note: Let > be the graph of the restriction 1 œ 0 lÐ! "Ó. For any set E © Ö!× ‚ Ò  "ß "Ó, a similar argument shows that > ∪ E is not path connected. In particular, cl > œ > ∪ ÐÖ!× ‚ Ò  "ß "ÓÑ is not path connected. But > is homeomorphic to Ð!ß "Ó, so > is path connected. So the closure of a path connected space need not be path connected. Definition 3.5 A space Ð\ß g Ñ is called a) locally connected if for each point B − \ and for each neighborhood R of B, there is a connected open set Y such that B − Y © R . b) locally path connected if for each point B − \ and for each neighborhood R of B, there is a path connected open set Y such that B − Y © R . Note: to say Y is path connected means that any two points in U can be joined by a path in U. Roughly, “locally path connected” means that “nearby points can be joined by short paths.” Example 3.6 1) ‘8 is connected, locally connected, path connected and locally path connected. 2) A locally path connected space is locally connected.

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3) Connectedness and path connected are “global” properties of a space \ : they are statements about \ “as a whole.” Local connectedness and local path connectedness are statement about what happens “locally” (in arbitrarily small neighborhoods of points) in \ In general, global properties do not imply local properties, nor vice-versa. a) Let \ œ Ð!ß "Ñ ∪ Ð"ß #Ó. \ is not connected (and therefore not path connected) but \ is locally path connected (and therefore locally connected). The same relations hold in a discrete space \ with more than one point. b) Let \ be the subset of ‘# pictured below. Note that \ contains the “topologist's sine curve” as a subspace  you need to imagine it continuing to oscillate faster and faster as it approaches the vertical line segment in the picture:

The \ is path connected (therefore connected, but \ is not locally connected: if : œ Ð!ß !Ñß there is no open connected set containing : inside the neighborhood R œ F " Ð:Ñ ∩ \ . Therefore \ is also not locally path connected. #

Notice that Examples a) and b) also show that neither “(path) connected” nor “locally (path) connected” implies the other.

Lemma 3.7 Suppose that 0 is a path in \ from + to , and 1 is a path from , to - . Then there exists a path 2 in \ from + to - . Proof 0 ends where 1 begins, so we feel intuitively that we can “join” the two paths “end-toend” to get a path 2 from + to - . The only technical detail to handle is that, by definition, a path 2 must be a function with domain Ò!ß "Ó. To get 2 we simply “join and reparametrize:”

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Define 2 À Ò!ß "Ó Ä \ by 2Ð>Ñ œ 

0 Ð#>Ñ ! Ÿ > Ÿ "# . (You can imagine a point moving 1Ð#>  "Ñ "# Ÿ > Ÿ " twice as fast as before: first along the path 0 and then continuing along the path 1.) The function 2 is continuous by the Pasting Lemma (see Exercise III.E22 ). ñ Theorem 3.8 If \ is connected and locally path connected, then \ is path connected. Proof If \ œ g, then \ is path connected. So assume \ Á g. For + − \ , let G œ ÖB − \ À there exists a path in \ from + to B×. Then G Á g since + − G (why?). We want to show that G œ \Þ Suppose B − G . Let 0 be a path in \ from + to B. Choose a path connected open set Y containing B. For any point C − Y , there is a path 1 in Y from B to C. By Lemma 3.7, there is a path 2 in \ from + to C, so C − GÞ Therefore B − Y © Gß so G is open. Suppose B  G and choose a path connected open set Y containing B. If C − Y , there is a path 1 in Y from C to B. Therefore there cannot exist a path in \ from + to C  or else, by Lemma 3.7, there would be a path 2 from + to B and B would be in G . Therefore C  G , so B − Y © \  G , so G is closed  and therefore clopen. \ is connected and G is a nonempty clopen set, so G œ \ . Therefore \ is path connected. ñ Here is another situation (particularly useful in complex analysis) where connectedness and path connected coincide: Corollary 3.9 An open connected set S in ‘8 is path connected.

Proof Suppose B − S. If R is any neighborhood of B in S , then B − intS R œ Y © S . Since S is open in ‘8 , and Y is open in S, Y is also open in ‘8 Þ Therefore there is an %  ! such that F% ÐBÑ © Y © R Þ Since F% ÐBÑ is an ordinary ball in ‘8 ß F% ÐBÑ is path connectedÞ (Of course,

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this might not be true for a ball in an arbitrary metric space.) We conclude that S is locally path connected so, by Theorem 3.8, S is path connected. ñ

4. Components Informally, the “components” of a space \ are its largest connected subspaces. A connected space \ has exactly one component  \ itself. In a totally disconnected space, for example , the components are the singletons ÖB×Þ In very simple examples, the components “look like” just what you imagine. In more complicated situations, some mild surprises can occur. Definition 4.1 A component G of a space \ is a maximal connected subspace. (Here, “maximal connected” means: G is connected and if G © H © \ where H is connected, , then G œ H)

For any : − \ , let G: œ ÖE À : − E © \ and E is connected×Þ Then Ö:× − G: ; since G: is a union of connected sets each containing :, G: is connected (Corollary 2.10). If G: © H and H is connected, then H was one of the sets E in the collection whose union defines G:  so H © G: and therefore G: œ HÞ Therefore G: is a component of \ that contains :, so can be written as the union of components: \ œ :−\ G: . Of course it can happen that G: œ G; when : Á ; À for example, in a connected space \ , G: œ \ for every : − \Þ But if G: Á G; , then G: ∩ G; œ g À if B − G: ∩ G; , then G: ∪ G; would be a connected set strictly larger than G: . The preceding paragraphs show that the distinct components of \ form a partition of \ À a pairwise disjoint collection whose union is \ . If we define : µ ; to mean that : and ; are in the same component of \ , then it is easy to see that “ µ ” is an equivalence relation on \ and that G: is the equivalence class of :. Theorem 4.2 \ is the union of its components. Distinct components of \ are disjoint and each component is a closed connected set. Proof In light of the preceding comments, we only need to show that each component G: is closed. But this is clear: G: © cl G: and cl G: is connected (Corollary 2.20). By maximality, we conclude that G œ cl G: Þ ñ

It should be clear that a homeomorphism maps components to components. homeomorphic spaces have the same number of components.

Therefore

Example 4.3 1) Let \ œ Ò"ß #Ó ∪ Ò$ß %Ó ∪ Ò&ß 'Ó © ‘Þ \ has three components: Ò"ß #Ó, Ò$ß %Ó, and Ò&ß 'Ó. For each ! Ÿ : Ÿ ", we have G: œ Ò!ß "ÓÞ If G is a component in a space \ that has only finitely many components, then \  G is the union of the other finitely many (closed) components. Therefore G is clopen.

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However, a space can have infinitely many components and in general they need not be open. For example, if \ œ Ö!× ∪ Ö 8" À 8 − × © ‘, then the components are the singleton sets ÖB× (why?). The component Ö!× is not open in \ . 2) In ‘# , \ œ #8œ" B"Î% ÐÐ8ß !ÑÑ is not homeomorphic to ] œ $8œ" B"Î% ÐÐ8ß !ÑÑ because \ has two components but ] has three. 3) If G © \ and G is nonempty connected and clopen, then G is a component of \ : for if g Á G © H © \ , then G is clopen in H so if H is connected, then G œ HÞ 4) The sets \ and ] in ‘# pictured below are not homeomorphic since \ contains a cut point : for which \  Ö:× has three components. ] contains no such cut point.

Example 4.4 The following examples are meant to help “fine-tune” your intuition about components by pointing out some false assumptions that you need to avoid. (Take a look back at Definition 2.4 to be sure you understand what is meant by a “disconnection.”) 1) Let \ œ Ö!× ∪ Ö 8" À 8 − ×Þ One of the components of \ is Ö!}, but Ö!× is not clopen in \ . Therefore the sets E œ Ö!× and F œ Ö 8" À 8 − × do not form a disconnection of \ . A component and its complement may not form a disconnection of \ Þ 2) If a space \ is the union of disjoint closed connected sets, these sets need not be components. For example, Ò!ß "Ó œ 