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P. 1 Math 490 Notes Connectedness Some Introductory Notes on P.147 p. 1 Math 490 notes Connectedness Some introductory notes on p.147 of Munkres remind us of the importance of connect- edness and compactness in analysis. These two notions, which form the subject matter of chapter 3 of Munkres, are often grouped together in topology books, not only because of their applications in analysis, but also because they exhibit some behavioral similarities. One thing they have in common is that they are topological properties which are defined not only for topological spaces, but also for subsets of topological spaces. Def N17.1 Let (X, τ) be a topological space, with A ⊆ X. A is disconnected iff there are open sets U, V in X such that: (1) U ∩ V ∩ A = φ, (2) U ∩ A = φ, (3) V ∩ A = φ, (4) A ⊆ U ∪ V . In this case, we say U and V disconnect or separate A.A connected subset is one which is not disconnected. A space (X, τ) is connected if X is a connected subset, as in Def N17.1. Thus (X, τ) is disconnected iff there are non-empty, disjoint open sets U and V whose union is X. In this case, U = X − V and V = X − U, so both U and V are clopen (simultaneously open and closed). Thus we have: Prop N17.1 A space (X, τ) is connected iff it has no clopen sets other than φ and X. It is a relatively simple matter to check that a subset A of a space (X, τ) is connected (as in Def N17.1) iff the subspace (A, τA) is connected. In any topological space, singleton sets and φ are connected; thus disconnected spaces can have connected subsets. A discrete space and all of its subsets other than φ and singletons are disconnected. An indiscrete space and all of its subsets are connected. p. 2 On p. 153, Munkres defines a linear continuum as a linearly ordered set L with more than one element such that (1) L has the least upper bound property, and (2) whenever x < y in L, there is a z ∈ L such that x<z<y. Munkres then proves (in Thm 24.1) that if L is a linear continuum, then in the order topology, all intervals, all rays, and L itself are connected. The proof of this fact is fairly straightforward, but somewhat lengthy, and we’ll omit it here. The intended point here is that since R (with the usual ordering) is a linear continuum, and the order topology corresponds to the usual topology, we may conclude that all intervals, all rays, and R itself are connected in the space (R, τu). Any set A in (R, τu) which is not an interval, ray or R is disconnected, since: A not an interval, ray or R ⇒ there are x, y ∈ A and z ∈ A such that x<z<y ⇒ (−∞, z) and (z, ∞) disconnect A. In summary, we have Prop N17.2: Prop N17.2 A subset of (R, τu) is connected iff it is an interval, ray, or R. Prop N17.3 If (X, τ) is connected and µ ≤ τ, then (X, µ) is connected. Proof : If(X, µ) is disconnected, then X contains a proper, nonempty µ-clopen subset which is also τ-clopen, so (X, τ) is also disconnected. ¥ Examples Consider our seven standard topologies on R: • τd and τs are both zero-dimensional, and any zero-dimensional space which is not indiscrete must be disconnected; • (R, τu) is connected by Prop N17.2, and since τcof, τo and τi are all coarser than τu, it follows by Prop N17.3 that (R, τcof), (R, τo) and (R, τi) are all connected; • (R, τcoc) is connected, since a cocountable subset of R can not have a cocountable comple- ment, so there are no τcoc-clopen sets in R except φ and R. p. 3 Prop N17.4 If f : (X, τ) → (Y, µ) is continuous and A ⊂ X is connected, then f(A) is connected. Proof : If there were U and V which disconnected f(A), then f −1(U) and f −1(V ) would disconnect A. ¥ Since connectedness is preserved under continuous maps, it is preserved under homeomor- phisms, and is therefore a topological property. An important consequence of Prop N17.4 is the Intermediate Value Theorem, which follows from Prop N17.4 and Thm 24.1 of Munkres. Intermediate Value Theorem (IVT) Let (X, τ) be a connected space, and let Y be a simply ordered set with order topology µ. Assume f : (X, τ) → (Y, µ) is continuous. Then if a and b are two points in X and r is a point in Y strictly between f(a) and f(b), there must exist a point c in X such that f(c)= r. Proof : Assuming all hypotheses, the sets U = f(X) ∩ (−∞, r) and V = f(X) ∩ (r, ∞) are disjoint, nonempty sets which are open in f(X). If there were no c in X such that f(c) = r, then we’d have f(X) = U ∪ V , so U and V would separate the space f(X). But f(X) must be connected, since it is the continuous image of a connected space. ¥ Note that if (X, τ) in the IVT is a linear continuum and a < b, then f : [a, b] → (Y, µ) is continuous, and [a, b] and f([a, b]) are connected, so the proof proceeds similarly with X replaced by [a, b]. The conclusion would then be that there must exist a c in [a, b] such that f(c)= r. Since c can not be a or b because f(c)= r and f(a) = r = f(b), we get the more familiar conclusion that c ∈ (a, b). p. 4 Corollary (the beginning calculus version of the IVT): If f : [a, b] → R is continuous and c is strictly between f(a) and f(b), there exists y ∈ (a, b) such that f(y)= c. Prop N17.5 If A is connected in (X, τ) and A ⊆ B ⊆ clτ A, then B is connected. In partic- ular, the closure of any connected set is connected. Proof : Assume A is connected in (X, τ) and A ⊆ B ⊆ clτ A. This proof is by contradiction, so suppose U and V disconnect B. Then U ∩ V ∩ B = φ, U ∩ B = φ, V ∩ B = φ, and B ⊆ U ∪ V . Since A ⊆ B, it clearly follows from the first and fourth of these conditions that U ∩V ∩A = φ and A ⊆ U ∪ V . If we can show that A ∩ U = φ and A ∩ V = φ, then U and V would separate A, which contradicts the assumption that A is connected. But we know there is an x ∈ U ∩ B, and since B ⊆ A , x ∈ U ∩ B ⇒ x ∈ U ∩ A ⇒ (x ∈ U and either x ∈ A or x is a limit point of A). Either way, it follows that U ∩A = φ. Similary, V ∩B = φ ⇒ V ∩A = φ. This contradiction shows that B is connected. ¥ Prop N17.6 If {Ai ¯ i ∈ I} is a collection of connected subsets in (X, τ) and there exists ¯ x ∈ \ Ai, then A = [ Ai is connected. i∈I i∈I Proof : Assume all hypotheses, and suppose U and V disconnect A. Then x (which is in A) must be in either U or V , but not both; say x ∈ U. Then we must have Ai ⊆ U for all i, since each Ai is connected, and Ai ⊆ U for some i ⇒ Ai ∩ V = φ ⇒ U, V disconnect Ai. Finally, since Ai ⊆ U for all i, we have A ⊆ U, and so A ∩ V = φ, a contradiction. Thus A can not be disconnected. ¥ p. 5 Let (X, τ) be a topological space with x ∈ X. Let Cx be the union of all connected subsets of X containing x. By Prop N17.6, Cx is connected, and it is clearly the largest connected subset containing x. Cx is called the connected component of x in (X, τ). Let P = {Cx ¯ x ∈ X} ¯ be the collection of all connected components in (X, τ). Prop N17.7 For any topological space (X, τ), P (as just defined) is a partition of X induced by the equivalence relation ∼ defined by: x ∼ y iff there exists a connected subset of X containing both x and y. Furthermore, each connected component Cx is closed. Proof : To prove That P is a partition of X, simply note that: (1) since x ∈ Cx for all x ∈ X, x = ∪P; (2) if z ∈ Cx ∩ Cy, then by Prop N17.6, Cx ∪ Cy is connected, and Cx = Cy = Cz. In checking that ∼ is an equivalence relation, reflexivity and symmetry should be clear. If x ∼ y and y ∼ z, then there exists connected sets A and B such that x, y ∈ A and y, z ∈ B. Then, by Prop N17.6, A ∪ B is connected and contains both x and z, so x ∼ z. To show that ∼ induces the partition P and vice-versa, note that: (1) x, y ∈ Cz for some z ⇒ x ∼ y; (2) x ∼ y ⇒ x, y both in some connected set A ⇒ A ⊆ Cx ∩ Cy ⇒ Cx ∩ Cy = φ ⇒ Cx = Cy (since P is a partition). Finally, for each x ∈ X, clτ Cx is connected by Prop N17.5, and since Cx is the largest con- nected set containing x, clτ Cx = Cx. ¥ If (X, τ) is connected, then for all x ∈ X, Cx = X. If X = R × R with the dictionary order topology, then for any a ∈ R, the sets Ua = {(x, y) ¯ x ≤ a} and Va = {(x, y) ¯ x>a} ¯ ¯ disconnect X.
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