Sketch the sets in $\mathbb C$ and determine whether they are open, closed, or neither; bounded; connected and their boundaries.

Question

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 1.27,8

Here are my answers:

1. Where have I gone wrong?
2. About connected, my topology is so far limited to the elementary topology in elementary analysis, complex analysis and real analysis, how exactly do we prove those sets are connected? Suppose they're not connected and then derive a contradiction like in my other question Elementary topology of $\mathbb C$: Union of 2 regions with nonempty intersection is a region ?
3. Is there a way to check this using computers? Not sure I can use Wolfram Alpha or guess I don't know how. Are there programs for this? Like can I do this in Matlab, scilab or R?

Answer 1

a) Wrong. It's the open disk $D(-3,2)$. In particular, it is open and it is not closed.

b) Right.

c) Right, but you should write $1$ instead of $(1,0)$.

d) Right, except that it is closed.

e) Right.

f) Right.

For each set, the easiest way to prove that it is connected consists in proving that it is path-connected.