Is the Set {z^2: z = x+iy, x>0, y>0} Open or Closed?

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In summary, the conversation discusses the notation used to represent a set of squares of complex numbers with positive real and imaginary parts. It is clarified that squaring z means taking the square of the complex number, and the set is defined as \{z\ :\ \exists x>0,\, y>0\ s.t.\ z = (x+iy)^2\} . The concept of open and closed sets is also discussed, with the conclusion that z \mapsto z^2 is a holomorphic mapping and its domain is open and connected. The importance of properly documenting the application of the open mapping theorem is also emphasized.
  • #1
sweetvirgogirl
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{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 don't know ... i need to know what squaring that z means
 
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  • #2
It's the set of squares of complex numbers with positive real and imaginary parts. Another way to write it would be:

[tex]\{z\ :\ \exists x>0,\, y>0\ s.t.\ z = (x+iy)^2\}[/tex]
 
  • #3
in that case ... wouldn't 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
sweetvirgogirl said:
in that case ... wouldn't 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)

well you've got strict inequalities everywhere...
 
  • #5
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
As for the openness/closedness, [tex]z \mapsto z^2[/tex] is a holomorphic mapping, and its domain is open and connected, so...
 
  • #7
Tantoblin said:
As for the openness/closedness, [tex]z \mapsto z^2[/tex] is a holomorphic mapping, and its domain is open and connected, so...
thats what i thought ... i wrote down "open" as my answer and the prof circles it and I don't 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
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
I think you'll have better luck looking at the function [tex]z \mapsto \sqrt{z}[/tex]. The preimage of an open set under a continuous function is open.
 
Last edited:

FAQ: Is the Set {z^2: z = x+iy, x>0, y>0} Open or Closed?

What is an openset?

An openset is a mathematical concept used in topology to describe a set of points that does not include its boundary. In other words, an openset is a set that does not contain any of its limit points.

What is a closedset?

A closedset is the complement of an openset, meaning that it includes its boundary points. In other words, a closedset is a set that contains all of its limit points.

What is the difference between an openset and a closedset?

The main difference between an openset and a closedset is that an openset does not include its boundary points, while a closedset does include its boundary points. This can also be seen as an openset being "open" and a closedset being "closed."

Can a set be both open and closed?

Yes, in some cases, a set can be both open and closed. This type of set is called a clopen set. An example of a clopen set is the set of all real numbers.

What does it mean for a set to be connected?

A set is said to be connected if it cannot be divided into two disjoint open sets. In other words, there are no gaps or breaks in the set. Intuitively, this means that all points in the set are "connected" in some way.

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