1. Definition 1.1.1. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. The set W is called open if, for every w 2 W , there is an > 0 such that B d (w; ) W . To show that X is In nitude of Prime Numbers 6 5. Connected spaces38 6.1. See the answer. Show transcribed image text. Show by example that the interior of Eneed not be connected. input point set. A set E X is said to be connected if E … The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. A set is said to be open in a metric space if it equals its interior (= ()). A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x n) < ε for any m ≥ n ε, n ≥ n ε. Theorem 2. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! Any unbounded set. Metric and Topological Spaces. We will consider topological spaces axiomatically. 10.3 Examples. if no point of A lies in the closure of B and no point of B lies in the closure of A. A metric space is just a set X equipped with a function d of two variables which measures the distance between points: d(x,y) is the distance between two points x and y in X. 1 If X is a metric space, then both ∅and X are open in X. Finite intersections of open sets are open. Set theory revisited70 11. 10 CHAPTER 9. a. Then S 2A U is open. Connected and Path Connected Metric Spaces Consider the following subsets of R: S = [ 1;0][[1;2] and T = [0;1]. Interlude II66 10. Complete Metric Spaces Deﬁnition 1. If each point of a space X has a connected neighborhood, then each connected component of X is open. This means that ∅is open in X. Paper 2, Section I 4E Metric and Topological Spaces One way of distinguishing between different topological spaces is to look at the way thay "split up into pieces". Now d(x;x 0) >0, and the ball B(x;r) is contained in U for every 0