if we defined

$\displaystyle \$

$\displaystyle A_r = ( (-\infty, a-r ] \cup [ a+r , \infty) \times \{b\} \cup ( (x,y) \in R^2 : (x-a)^2 + (y-b)^2 = r^2 , y\leq b ) $

prove that this is homeomorphic with R I draw A_r in the picture that I attached

in latex Prove

$\displaystyle (A_r,t_u ) \cong (R, t_u) $

as I said $\displaystyle t_u $ is the usual topology

what I came up is if I can prove that $\displaystyle (a-r , a+r) \cong ((x,y) \in R^2 : (x-a)^2 + (y-b)^2 = r^2 , y\leq b ) $

since

$\displaystyle ( -\infty , a-r] \times {b} \cong (-\infty , a-r] $ two spaces with usual topology

$\displaystyle [a+r , \infty) \times {b} \cong [a+r ,\infty) $ two spaces with usual topology

so the the problem is how I can prove the semi-circle (x-a)^2 + (y-b)^2 y>=b is homemorphic with the interval (a-r , a+r) or can it be, my prof said

$\displaystyle (A_r, t_u) \cong (R,t_u) $

but it is not clear how

that's my question and here is A_r after I changed something

or instead of that prove that $\displaystyle A_r $ is connected space