Define A = {(x,y) in R^2 : x^2+y^2 <= 4 and y >= x} U {(2,2)}.
a) Is A open?
b) Is A closed?
c) What is int A?
d) What is ext A?
e) What is bd A?
f) Is A bounded?
g) Is A a compact subset of R^2?
No, there are points in A for which every neighbourhood of the point contains
points not in the set.
Yes, it contains all its limit pointsb) Is A closed?
(check these from here on)c) What is int A?
{(x,y): x^2+y^2<4, and y>x}
{(x,y): x^2+y^2>4, or y<x}d) What is ext A?
{(x,y): x^2+y^2=4, and y>x}U {(x,y): y=x and x in [-sqrt(2),sqrt(2)]} U {(2,2)}e) What is bd A?
yesf) Is A bounded?
It is closed and bounded and so compact.g) Is A a compact subset of R^2?
RonL