open sets

[triclino]

Prove that (0,2) is open in [math](R^2,d)[/math] where d is the discrete metric:

 

 

.....................................................d(x,y) = 0 ,if x=y...........................................................

 

......................................................d(x,y) =1 ,if [math]x\neq y[/math].....................................

 

Obviously we have to prove that:

 

for all xe(0,2) ,there exists ε>0 such that B(x,ε) is a subset of (0,2) ,or

 

for all xε(0,2) ,there exists ε>ο such that yεB(x,y) iplies [math]0<y_{1}<2[/math],where x =[math](x_{1},0)[/math] and y=[math](y_{1},y_{2})[/math]

 

AM I correct??

Edited October 14, 2011 by triclino