Question

Determine the interior, the boundary and the closure of the set {z ε: Re(z^2>1}
Is the interior of the set path-connected?

Answer 1

Alright so z^2=(x+iy)(x+iy)=x^2+2ixy-y^2

so Re(x^2+2ixy-y^2)= x^2-y^2 >1

So would the image be a hyperbola that starts when each axis is >1 leaving a hole in the center?

Boundary: {z ε: Re(z^2)=1}
Interior: none
Closure: {z ε: Re(z^2)>1}

Since there is no interior the question of interior path connectedness is mout.

I'm not sure if this is right though
@eliassaab @experimentX @UnkleRhaukus @phi @eseidl

Answer 2

Wouldn't the interior be R(^2) > 1 because for there exist an r > 0 such that
$$
B_r(x)\subset R(z^2)> 1
$$
for all x in R(z^2) > 1?

Answer 3

Boundary looks right.

Answer 4

That helps! Thanks!

Answer 5

you're welcome