Prove that

is a connected space, t_u is the usual topology

defined

A_r = ( (-infinity , a-r ] U [ a+r , infinity ) ) X {b} U {(x,y) in R^2 : (x-a)^2 + (y-b)^2 =r^2 y>= b }

B_r = ( (-infinity , a-r ] U [ a+r , infinity ) ) X {b} U {(x,y) in R^2 : (x-a)^2 + (y-b)^2 =r^2 y<= b }

A_r homomorphic with R with usual topology , thats my question I do not know why A_r homorophic with R^1 ???

A_r connected

B_r connected .... I get all the rest it is easy

here is a picture that shows A_r and B_r