{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture1

lecture1 - Lecture 1 1 1.1 Review of Topology Metric Spaces...

This preview shows pages 1–2. Sign up to view the full content.

Lecture 1 1 Review of Topology 1.1 Metric Spaces Definition 1.1. Let X be a set. Define the Cartesian product X × X = { ( x, y ) : x, y X } . Definition 1.2. Let d : X × X R be a mapping. The mapping d is a metric on X if the following four conditions hold for all x, y, z X : (i) d ( x, y ) = d ( y, x ), (ii) d ( x, y ) 0, (iii) d ( x, y ) = 0 ⇐⇒ x = y , and (iv) d ( x, z ) d ( x, y ) + d ( y, z ). Given a metric d on X , the pair ( X, d ) is called a metric space . Suppose d is a metric on X and that Y X . Then there is an automatic metric d Y on Y defined by restricting d to the subspace Y × Y , d Y = d Y × Y. (1.1) | Together with Y , the metric d Y defines the automatic metric space ( Y, d Y ). 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. We review open sets, closed sets, norms, continuity, and closure. Throughout this section, we let ( X, d ) be a metric space unless otherwise specified. One of the basic notions of topology is that of the open set. To define an open set, we first define the -neighborhood. Definition 1.3. Given a point x o X , and a real number � > 0, we define U ( x o , � ) = { x X : d ( x, x o ) < � } . (1.2) We call U ( x o , � ) the -neighborhood of x o in X . Given a subset Y X , the -neighborhood of x o in Y is just U ( x o , � ) Y . Definition 1.4. A subset U of X is open if for every x o U there exists a real number � > 0 such that U ( x o , � ) U .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

lecture1 - Lecture 1 1 1.1 Review of Topology Metric Spaces...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online