1. In this exercise you will prove that disc {(x, y): + y < 1}...

Transcribed Image Text:

1. In this exercise you will prove that disc {(x, y): ² + y² < 1} is an open subset of R2, and then that every open disc in the plane is an open set. : (i) Let (a, b) be any point in the disc D = {(x,y) x² + y² < 1}. Put T= a²+6². Let Rab) be the open rectangle with vertices at the points (ar, b± ¹r). Verify that R(a,b) CD. (II) Using (1) show that D = UR(a,b) (a,b) ED (iii) Deduce from (ii) that D is an open set in R². (iv) Show that every disc {(x, y): (-a)² + (y-b)² < c², a, b, c = R} is open in R². 1. In this exercise you will prove that disc {(x, y): ² + y² < 1} is an open subset of R2, and then that every open disc in the plane is an open set. : (i) Let (a, b) be any point in the disc D = {(x,y) x² + y² < 1}. Put T= a²+6². Let Rab) be the open rectangle with vertices at the points (ar, b± ¹r). Verify that R(a,b) CD. (II) Using (1) show that D = UR(a,b) (a,b) ED (iii) Deduce from (ii) that D is an open set in R². (iv) Show that every disc {(x, y): (-a)² + (y-b)² < c², a, b, c = R} is open in R².