1. ## Topology Proof?

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 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??

2. What are you expected to do to show a set is open?
I ask because 1b is often used as a basic open set.
Therefore, you must be using some other definition.

3. hmm well my definition for an open set is :

the set S is said to be open if intS = S (int = interior)

is this what you had in mind??

how about 2a.. have i done that bit correctly?

4. Originally Posted by heyo12
my definition for an open set is :
the set S is said to be open if intS = S (int = interior)
The statement that $\displaystyle z_0 \in {\mathop{\rm int}} (S)$ means that $\displaystyle \left( {\exists r > 0} \right)\left[ {\left\{ {z:\left| {z - z_0 } \right| < r} \right\} \subset S} \right]$.

Then for #1a, choose $\displaystyle r = \frac{{{\mathop{\rm Re}\nolimits} (z_0 )}}{2}$.

This is a Post Script.
For 1b, by definition an open disk is its own interior.