Thread: Compact subset problem

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?

Originally Posted by tttcomrader

Define A = {(x,y) in R^2 : x^2+y^2 <= 4 and y >= x} U {(2,2)}.

a) Is A open?

No, there are points in A for which every neighbourhood of the point contains
points not in the set.

b) Is A closed?

Yes, it contains all its limit points

c) What is int A?

(check these from here on)

{(x,y): x^2+y^2<4, and y>x}

d) What is ext A?

{(x,y): x^2+y^2>4, or y<x}

e) What is bd 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)}

f) Is A bounded?

yes

g) Is A a compact subset of R^2?

It is closed and bounded and so compact.

RonL