# Thread: Open and Closed Sets (Complex Analysis)

1. ## Open and Closed Sets (Complex Analysis)

(a) Describe the interior and the boundary
(b) State whether the set is open or closed or neither open nor closed
(c) State whether the interior of the set is connected (if it has an interior)

1) $\displaystyle A =${$\displaystyle z: |z| < 1$ or $\displaystyle |z-3| \leq4$}

2) $\displaystyle B =${$\displaystyle z: Re(z^2) = 4$}

I'll try to tackle #2.

2)
$\displaystyle z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy$
$\displaystyle x^2-y^2 = 4$
If x=2, then y=0
So...
(a) int(B) = none? Boundary is ?
(b) B is closed.
(c) It is connected?

2. The equation x^2 - y^2 = 4 defines a hyperbola. As $\displaystyle x \to \infty$ then $\displaystyle y \to \infty$ so it's defly not closed.

It's not even connected because it's a hyperbola and that's got 2 bits to it.

There is no interior because the set consists of the line, and so it's all boundary.

I think. I need to brush up my interpretation of terminology.