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