Example 7.4. The set of real numbers R with the function d(x;y) = jx yjis a metric space. Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. In other words, no sequence may converge to two diﬀerent limits. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Remarks. De¿nition 5.1.10 Suppose that A is a subset of a metric space S˛dS and that f is a function with domain A and range contained in a metric space X˛dX ˚ i.e., f : A ˆ X. Results E is closed if every limit point … 1.3 Closed Sets (in a metric space) While we can and will deﬁne a closed sets by using the deﬁnition of open sets, we ﬁrst deﬁne it using the notion of a limit point. [1,2]. Limit points De nition { Limit point Let (X;T) be a topological space and let AˆX. Proof. Note. If x 2E and x is not a limit point of E, then x is called anisolated pointof E. E is dense in X if every point of X is a limit point of E, or a point of E (or both). A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Defn A sequence {x n} in a metric space (X,d) is said to converge, to a point x 0 say, if for each neighborhood of x 0 there exists a natural number N so that x n belongs to the neighborhood if n is greater or equal to N; that is, eventually the sequence is contained in the neighborhood.In this case, we say that x 0 is the limit of the sequence and write This is the most common version of the definition -- though there are others. Theorem 1.3 – Limits are unique The limit of a sequence in a metric space is unique. We say that xis a limit point of Aif every neighbourhood of xintersects Aat a point other than x. Theorem 2.7 { Limit points and closure Let (X;T) be a topological space and let AˆX. Deﬁnition 1.3.1. So 0 ∈ A is a point of closure and a limit point but not an element of A, and the points in (1,2] ⊂ A are points of closure and limit points. Limit points and closed sets in metric spaces. A subset Uof a metric space Xis closed if the complement XnUis open. Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. Let (X;d) be a metric space and E ˆX. Limit points are also called accumulation points. Definition If A is a subset of a metric space X then x is a limit point of A if it is the limit of an eventually non-constant sequence (a i) of points of A. Theorem 17.6 Let A be a subset of the topological space X. Examples Sequential Convergence. The following result gives a relationship between the closure of a set and its limit points. Limit points are also called accumulation points of Sor cluster points of S. Example 1. A point z is a limit point for a set A if every open set U containing z intersects A in a point other than z. Recall that, in a metric space, compactness, limit point compactness, and sequential compactness are equivalent (see Theorem 28.2). De nition: A limit point of a set Sin a metric space (X;d) is an element x2Xfor which there is a sequence in Snfxgthat converges to x| i.e., a sequence in S, none of whose terms is x, that converges to x. A point x2Xis a limit point of Uif every non-empty neighbourhood of x contains a point of U:(This de nition di ers from that given in Munkres). The set Uis the collection of all limit points of U: By a neighbourhood of a point, we mean an open set containing that point. Lemma 43.1 states that a metric space in complete if every Cauchy sequence in X has a convergent subsequence. 94 7. The Topology of Metric Spaces ... Deﬁnition 9.3 Let (X,C)be a topological space, and A⊂X.Then x∈Xis called a limit point of the set Aprovided every open set Ocontaining xalso contains at least one point a∈A,witha=x. 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