Chapter V Connected Spaces

1 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 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, E F where E ! and F ! . 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 \ !

2 As disconnected. Here, we can write \ E F , where E ( ! . and F ! . 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) F cl\ E E cl\ F 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 Suppose E and F are subspaces of \ g . E and F are called separated if each is disjoint from the closure of the other that is, if F cl\ E E cl\ F g. It follows immediately from the definition that i) separated sets must be disjoint, and ii) subsets of separated sets are separated: if E F are separated, G E and H F , then G and H are also separated. Example 1) In , the sets E ! and F ! are disjoint but not separated. Likewise in # , the sets E B C B# C# " and F B C B 2 # C# " are disjoint but not separated.

3 213. 2) The intervals E ! and F ! are separated in but cl E cl F g . The same is true for the open balls E B C B# C# " and F B C B 2 # 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 In any space \ g , the following statements are equivalent: 1) g and \ are the only clopen sets in \. 2) if E \ and Fr E g, then E g or E \. 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 Fr E 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 \ E are pieces of \ that are separated from each other. Proof 1) 2) This follows because E is clopen iff Fr E g (see Theorem ).

4 1) 3) Suppose 3) is false and that \ E F where E, F are disjoint, nonempty and open. Since \ E F is open and nonempty, we have that E is a nonempty proper clopen set in \ , which shows that 1) is false. 3) 4) This is clear. 4) 5) If 5) is false, then \ E F , where E F are nonempty and separated. Since cl F E g we conclude that cl F F , so F is closed. Similarly, E must be closed. Therefore 4) is false. 5) 1) Suppose 1) is false and that E is a nonempty proper clopen subset of \ . Then F \ E is nonempty and clopen, so E and F are separated. Since \ E F , 5) is false.. Definition A space \ g is Connected if any (therefore all) of the conditions 1) - 5) in Theorem 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 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.

5 3) E and F are nonempty and separated in G . If G is disconnected, such a pair of sets E 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. 214. Theorem Suppose E F G \ Then E and F are separated in G iff E and F are separated in \ . Proof clG F G cl\ F (see Theorem ), so E clG F g iff E cl \ F G g iff E G cl \ F g iff E cl \ F g Similarly, F clG E g iff F cl\ E g . Caution: According to Theorem , G is disconnected iff G E F where E and F are nonempty separated set in G iff G E F where E and F are nonempty separated set in \ . Theorem 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 E F in G, then phrase in G cannot be omitted: the sets E, F might not be open (or closed) in \.

6 For example, suppose \ ! " and G ! "# "# " . The sets E ! "# and ". F # " are open, closed and separated in G . By Theorem , they are also separated in . but they are neither open nor closed in . Example 1) Clearly, connectedness is a topological property. More generally, suppose 0 \ ]. is continuous and onto. If F is proper nonempty clopen set in ] , then 0 " F 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 ". In particular, and are not Connected . 3) is not Connected since we can write as the union of two nonempty separated sets: ; ; # # ; ; # # . Similarly, we can show is not Connected . More generally suppose G and that G is not an interval. Then there are points + D , where + , G but D G Then B G B D B G B 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.

7 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 ) using a different argument. 4) (The Intermediate Value Theorem) If \ is Connected and 0 \ is continuous, then ran 0 is Connected (by part 1) so ran 0 is an interval (by part 3). Therefore if + , \. and 0 + D 0 , , there must be a point - \ for which 0 - D . 5) The Cantor set G is not Connected (since it is not an interval). But much more is true. Suppose B C E G and that B C Since G is nowhere dense (see ), the interval B C . G , so we can choose D G with B D C. Then F D E. D 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 \ g is called totally disconnected every Connected subset E satisfies lEl " The Spaces and are other examples of totally disconnected Spaces .

8 6) \ is Connected iff every continuous 0 \ ! " is constant: certainly, if 0 is continuous and not constant, then 0 " ! is a proper nonempty clopen set in \ so \ is not 215. Connected . Conversely, if \ is not Connected and E is a proper nonempty clopen set, then the characteristic function ;E \ ! " is continuous but not constant. Theorem Suppose 0 \ ] . Let > B C \ ] C 0 B the graph of 0 . 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 \ > be defined by 2 B B 0 B Clearly 2 is a one-to-one map from \ onto >. Let + \ and suppose Y Z > is a basic open set containing 2 + + 0 + . Since 0 is continuous and 0 + Z there exists an open set S in \ containing + and such that 0 S Z Then + Y S, and 2 Y S Y Z >, so 2 is continuous at +. If Y is open in \ , then 2 Y Y ] > is open in >, so 2 is open.

9 Therefore 2 is a homeomorphism.. Note: It is not true that a function 0 with a Connected graph must be continuous. See Example The following lemma makes a simple but very useful observation. Lemma Suppose Q 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 G Q G R . But G is Connected so G Q and G R cannot form a disconnection of G . Therefore either G Q g (so G R or G R g (so G Q ).. 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 . and G are not separated. Then W G G is Connected . Theorem Suppose G and G ( M ) are Connected subsets of \ and that for each , G . Proof Suppose that W Q R where Q and R are separated.)

10 By Lemma , either G Q. or G R . Without loss of generality, assume G Q . By the same reasoning we conclude that for each , either G Q or G R But if some G R , then G and G would be separated. Hence every G Q . Therefore R g and the pair Q R is not a disconnection of W.. " M , C G" g. Then C M is Connected . Corollary Suppose that for each M , C is a Connected subset of \ and that for all 216. Proof If M g then C M g is Connected . If M g, pick an ! M and let G ! be separated. By Theorem , C M is Connected .. the central set G in Theorem For all M , G G ! g, so G and G ! are not Then . Corollary For each 8 , suppose G8 is a Connected subset of \ and that G8 Gn " g. 8 " G8 is Connected . Proof Let E8 85 " G5 . Corollary (and simple induction) shows that the E8 's are that . Connected . Then g E" E# E8 Another application of Corollary gives 8 " E8 8 " G8 is Connected .. Corollary Let M Then M is Connected iff M is an interval. In particular, is Connected , so and g are the only clopen sets in.