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A topological space X is path-connected if every pair of points is connected by a path. However, we can prove the following result about the canonical map ˇ: X!X=˘introduced in the last section. Theorem 26. Example 4. Give ve topologies on a 3-point set. De ne whatit meansfor a topological space X to be(i) connected (ii) path-connected . Every path-connected space is connected. At this point, the quotient topology is a somewhat mysterious object. The topology … A topological space (X;T) is path-connected if, given any two points x;y2X, there exists a continuous function : [0;1] !Xwith (0) = x and (1) = y. We will allow shapes to be changed, but without tearing them. (In other words, if f : X → Y is a continuous map and X is connected, then f(X) is also connected.) X is connected if it has no separation. Proof. Connectedness. METRIC AND TOPOLOGICAL SPACES 3 1. The number of connected components is a topological in-variant. 11.N. called connected. The idea of a topological space. A separation of a topological space X is a partition X = U [_ W into two non-empty, open subsets. 1 x2A ()every neighbourhood of xintersects A. Consider the interval [0;1] as a topological space with the topology induced by the Euclidean metric. Let ˝ A be the collection of all subsets of Athat are of the form V \Afor V 2˝. topological space Xwith topology :An open set is a member of : Exercise 2.1 : Describe all topologies on a 2-point set. (Path-connected spaces.) Suppose (X;T) is a topological space and let AˆX. There is also a counterpart of De nition B for topological spaces. Just knowing the open sets in a topological space can make the space itself seem rather inscrutable. Topology underlies all of analysis, and especially certain large spaces such (It is a straightforward exercise to verify that the topological space axioms are satis ed.) Connectedness is a topological property. The property we want to maintain in a topological space is that of nearness. [You may assume the interval [0;1] is connected.] The image of a connected space under a continuous map is connected. Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. 11.O Corollary. Definition. 11.Q. Proposition 3.3. Let Xbe a topological space with topology ˝, and let Abe a subset of X. 1 Connected and path-connected topological spaces De nition 1.1. Give a counterexample (without justi cation) to the conver se statement. Then ˝ A is a topology on the set A. A topological space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0,1] to X such that f(0) = x and f(1) = y (This function is called a path from x to y). If A is a P β-connected subset of a topological space X, then P β Cl (A) is P β-connected. 11.P Corollary. Recall that a path in a topological space X is a continuous map f:[a,b] → X, where[a,b]⊂Ris a closed interval. Exercise 2.2 : Let (X;) be a topological space and let Ube a subset of X:Suppose for every x2U there exists U x 2 such that x2U x U: Show that Ubelongs to : In other words, we have x=2A x=2Cfor some closed set Cthat contains A: Setting U= X Cfor convenience, we conclude that x=2A x2Ufor some open set Ucontained in X A R with the standard topology is connected. This will be codi ed by open sets. Prove that any path-connected space X is connected. A continuous image of a connected space is connected. By de nition, the closure Ais the intersection of all closed sets that contain A. Definition. The discrete topology is clearly disconnected as long as it contains at least two elements. , we can prove the following result about the canonical map ˇ:!. 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