First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces.Further it covers metric spaces, Continuity and open sets for metric spaces, Closed sets for metric spaces, Topological spaces, Interior and closure, More on topological structures, Hausdorff spaces and Compactness. The main ideas … Because of this, the ﬁrst third of the course presents a rapid overview of metric spaces (either as revision or a ﬁrst glimpse) to set the scene for the main topic of topological spaces. Students will gain knowledge of definitions, theorems and calculations in • Normed, Metric and Topological spaces • Open and closed sets and their relation to continuity I will be making typeset notes available every 2 weeks. First, a reminder. A Note from the Author. Please note, the full solutions are only available to lecturers. Question 6: Give the de nition of the diameter of a subset of a metric space. A set X equipped with a function d: X X !R 0 is called a metric space (and the function da metric or distance function) provided the following holds. It has a very strong grounding in making sure that clear explanations and meaningful carefully laid out proofs are given throughout the book. Limit in the Cartesian Product § 7. 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. Answer: A topological space is said to be 1. compact if every open cover possesses a nite subcover, and 2. connected if it admits no nontrivial partition into open sets. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. Having established this firm foundation the author generalises these concepts to metric spaces, talking about open and closed sets, interiors and boundaries. The main ideas of … Complete Metric Spaces and Function Spaces. METRIC AND TOPOLOGICAL SPACES 3 1. 3) analyse and synthesise the properties of a number of standard classes of metric spaces (for example Euclidean spaces, function spaces, etc.). (2 credits) Answer: The diameter of a subset Aof a metric space (X;d) is supfd(x;y) j(x;y) 2 A Ag. tion for metric spaces, a concept somewhere halfway between Euclidean spaces and general topological spaces. Course Objectives: The purpose of this class is to introduce the notion of topological spaces, ﬂrst for metric spaces and then more generally. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. To introduce the notions of Normed Space, Metric Space and Topological Space, and the fundamental properties of Compactness, Connectedness and Completeness that they may possess. Uniform Convergence Exercises X. Topological Spaces § 1. We call these structures metric spaces, and their properties, when further generalized, give rise to the elements of topological spaces and the discipline of Topology. Metric Spaces § 2. geometry has been an inseparable part of mathematics. About this website. 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 … An Introduction to Metric and Topological Spaces (Second Edition) Wilson A. Sutherland OXFORD UNIVERSITY PRESS 2009, 224 PAGES PRICE (HARDBACK) £45.00 ISBN 978-0-19-956307-4 A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and …