This preview shows page 1. Sign up to view the full content.
Unformatted text preview: iv. Therefore ( x ) = ( z ) is an open neighborhood of x such that x z for every x ( x ) Similarly, ( z ) = ( x ) is an open neighborhood of z such that z x for every z ( z ). Consequently x z for every x ( x ) and z ( z ) 3. ( y ) = ( ( y ) ) (Exercise 1.56). Therefore, ( y ) is closed if and only if ( y ) is open (Exercise 1.80). Similarly, ( y ) is closed if and only if ( y ) is open. 1.237 1. Let = { ( x , y ) : x y } . Let (( x , y )) be a sequence in which converges to ( x , y ). Since ( x , y ) , x y for every . By assumption, x y . Therefore, ( x , y ) which establishes that is closed (Exercise 1.106) Conversely, assume that is closed and let (( x , y )) be a sequence converging to ( x , y ) with x y for every . Then (( x , y )) which implies that...
View Full
Document
 Fall '10
 Dr.DuMond
 Macroeconomics

Click to edit the document details