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