Proving Open and Closed Sets: A How-to Guide

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To prove that the left half-plane {z: Re z > 0} is open, one can show that for any point z in this set, there exists a disk around z that remains entirely within the set, utilizing the definition of an open set. For the open disk D(z0, r), a similar approach applies; for any point p in the disk, a smaller disk can be found that is also contained within D(z0, r), which can be demonstrated using the triangle inequality. To prove that the closed disk D(z0, r) is closed, one must show that its complement is open, meaning that for any point outside the closed disk, there exists a neighborhood around it that does not intersect the closed disk. The discussion highlights the importance of correctly applying definitions and properties of open and closed sets in complex analysis. Understanding these concepts is crucial for accurately proving the nature of various sets in mathematical contexts.
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How can you prove sets
1---------
how can u prove the following sets are are open,
a. the left half place {z: Re z > 0 };
b. the open disk D(z0,r) for any [math]z_0 \varepsilon C[/math] and r > 0.

2---------
a. how can u prove the following set is a closed set:
_
D(z0, r)


MY WORKING SO FAR
1.. could you please give me a hint on how to start a and b as I've researched but still haven't got much of an idea. once i get a little hint then ill try solving and show you my working..

2a.
--------
if D(z0,r) is closed, this implies C\S (the compliment) is open. Therefore, for any z not belonging to the set, there is an e > 0 such that D(z,e) C C\S. This further implies z is not a limit point of S which means that it is a closed set?

is this correct proof for 2a??
 
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heyo12 said:
How can you prove sets
1---------
how can u prove the following sets are are open,
a. the left half place {z: Re z > 0 };
b. the open disk D(z0,r) for any z_0 \varepsilon C] and r > 0.
Use the definition of open set, of course. If the real part of z is negative, can you find a disk about z such that every point in it has real part negative? (b) is a little harder. Show that for any point p in D(z0,r) there exist a disk about p that is a subset of D(z0,r). You will need the triangle inequality.

2---------
a. how can u prove the following set is a closed set:
_
D(z0, r)


MY WORKING SO FAR
1.. could you please give me a hint on how to start a and b as I've researched but still haven't got much of an idea. once i get a little hint then ill try solving and show you my working..

2a.
--------
if D(z0,r) is closed, this implies C\S (the compliment) is open. Therefore, for any z not belonging to the set, there is an e > 0 such that D(z,e) C C\S. This further implies z is not a limit point of S which means that it is a closed set?

is this correct proof for 2a??
No, you certainly cannot start a proof That \overline{D(z0,r)} is closed by saying "if \overline{D(z0,r)} is closed"!
What, exactly, is your definition of \overline{D(z0,r)}?
 
well my definition of \overline{D}(z0,r)} was that it is a set which is a closed disk??

\overline{D(z0,r)} { w: |z0 - w | < r }
 

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