Can someone explain these two examples of an open set

[Austin Chang]

Screenshot below
-Thanks

 

[FactChecker]

4) Apparently they have already proven that the open ball in R is an open set. That may be their original definition of an open set. They have also proven that any union of open sets is open. So all they need to do is to rewrite the original set of 4) as the union of balls.
5) Same logic as in 4) except that the balls are in the complex plane, C.

 

[WWGD]

Essentially a set U is open if for every x in U there is an open set ( possibly an open ball) S so that x is an element of S and S is fully contained in U . Can you take it from there @Austin Chang ?

 

[PeroK]

Essentially a set U is open if for every x in U there is an open set ( possibly an open ball) S so that x is an element of S and S is fully contained in U . Can you take it from there @Austin Chang ?

You seem to be going round resurrecting a lot of old threads. This one is from 2016.

 

[WWGD]

You seem to be going round resurrecting a lot of old threads. This one is from 2016.

I didn't notice the dates. I guess they are still open though. I am using my phone, which does not give me the same access as PC.Edit : I will ask staff if it is ok.

 

[WWGD]

@PeroK , I just spoke with Greg, he seems to be ok with it. You may chime in if you wish.