Proving Open and Closed Sets: A How-to Guide

[heyo12]

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 \(\displaystyle z_0 \varepsilon C\) 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??

 

[HallsofIvy]

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 $\overline{D(z0,r)}$?

 

[heyo12]

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 }