shouldn't it be homeomorphic instead of homomorphic?
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
if we defined
prove that this is homeomorphic with R I draw A_r in the picture that I attached
in latex Prove
as I said is the usual topology
what I came up is if I can prove that
since
two spaces with usual topology
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
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 is connected space