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