How can you prove sets

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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 and r > 0.

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a. how can u prove the following set is a closed set:

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D(z0, r)

MY WORKING SO FAR

1.. could you please give me a hint on how to start a and b as ive researched but still havent got much of an idea. once i get a little hint then ill try solving and show you my working..

2a.

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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??