# Openset / closedset / connected

1. May 3, 2006

### sweetvirgogirl

{z^2: z = x+iy, x>0, y>0}

i am a lil confused about the notation to represent the set ...

i'm used to seeing {z: z = x+iy, x>0, y>0}
but what effect does squaring z have?

i thought the set was open simply because x>0 and y>0 ... but apprently i was wrong ... (or maybe not?) ... i dunno ... i need to know what squaring that z means

2. May 4, 2006

### AKG

It's the set of squares of complex numbers with positive real and imaginary parts. Another way to write it would be:

$$\{z\ :\ \exists x>0,\, y>0\ s.t.\ z = (x+iy)^2\}$$

3. May 4, 2006

### sweetvirgogirl

in that case ... wouldnt it be an open set?
and it will be above real axis? (meaning the boundary is upper plane or lower plane? getting confused with terminology a little)

4. May 4, 2006

### fourier jr

well you've got strict inequalities everywhere...

5. May 4, 2006

### AKG

The boundary is just the real line. Note it's usually good to distinguish "strict upper half plane" and "non-strict upper half plane" so you don't confuse yourself or others.

6. May 6, 2006

### Tantoblin

As for the openness/closedness, $$z \mapsto z^2$$ is a holomorphic mapping, and its domain is open and connected, so...

7. May 9, 2006

### sweetvirgogirl

thats what i thought .... i wrote down "open" as my answer and the prof circles it and I dont think I got any points for it ... yes, i didnt write connected, but I should at least get half the points or something. oh well maybe he didnt gimme any credit, because I didnt explain why I think it's open set

8. May 10, 2006

### Tantoblin

Yes, well the crucial point here is that you are applying the open mapping theorem, which works only when a number of conditions are satisfied. The open mapping theorem is very nontrivial and even counterintuitive, so you should properly document its application.

9. May 17, 2006

### Jimmy Snyder

I think you'll have better luck looking at the function $$z \mapsto \sqrt{z}$$. The preimage of an open set under a continuous function is open.

Last edited: May 17, 2006