Open and closed sets in metric space pdf

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4 Closed and Open Sets. We now know the general framework in which we can study convergence of iterative processes. We have to have a set in which we do the iteration and a way to measure the distance between two points in the set. That is, we need a metric space. We also know what it means for a sequence to 13 Aug 2015 A set is open if at any point we can find a neighborhood of that point contained in the set. Let (X, d) be a metric space. A set A ? X is open if ?x ? A ?? > 0 such that B?(x) ? A Remember that B?(x) = {y ? X : d(y, x) < ?} so openness depends on X. A set C ? X is closed if X \ C is open. Open and Closed Sets. Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points — i.e., if. Vx ? S : 9? > 0 : B(x, ?) C S. (1). Theorem: (O1) ? and X are open sets. (O2) If S1,S2,,Sn are open sets, then nn i=1Si is an open set. (O3) Let A be an arbitrary set. If S? is an open set for Also included are some theorems due to Baire. 2.1. Open and Closed Sets. There are special types of sets that play a distinguished role in analysis; these are the open and closed sets. To expedite the discussion, it is helpful to have the notion of a neighbourhood in metric spaces. Definition 2.1.1. Let (X, d) be a metric space If U is an open subset of a metric space (X, d), then its complement Uc = X - U is said to be closed. In other words, a set is closed if and only if its complement is open. For example, a moments thought should convince you that the subset of ®2 defined by {(x, y) ? ®2: x2 + y2 ? 1} is a closed set. The closed ball of radius r Corollary 1.6. Let A be a subset of a metric space X. Then A is closed and is contained inside of any closed subset of X which contains A. Proof. Since the complement of A is equal to int(X ? A), which we know to be open, it follows that A is closed. If Z is any closed set containing A, we want to prove that Z contains A (so A is. 4 Open sets and closed sets. Throughout this section, (X, ?) is a metric space. Definition 4.1. A set A ? X is open if it contains an open ball about each of its points. That is, for all x ? A, there exists ? > 0 such that B?(x) ? A. Lemma 4.2. An open ball in a metric space (X, ?) is an open set. Proof. If x ? Br(?) then ?(x, ?) = r ? ? In any a metric space (X, d), the limit of a convergent sequence is unique. Proof. Suppose that xn > x and xn > x , where x = x ; then d := d(x, x ) > 0, and ? may be chosen as d/2. So B?(x) and B?(x ) are disjoint, by the triangle inequality. Definition 2. Let X be a metric space. All points and sets mentioned below and understood to be elements and subsets of X. i) If x ? Rk and r > 0, the open (or closed) ball B with center x and radius r is defined to be the set of all y ? Rk such that y ? x < r. ii) A neighborhood of p is a set Nr(p) consisting of all q such that d(p,