Can someone explain these two examples of an open set

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In summary, the conversation is discussing the definition of an open set and how it applies to different sets, specifically in R and the complex plane C. The main point is that a set is considered open if for every element x in the set, there is another open set or ball S that contains x and is fully contained within the original set. The conversation also mentions the importance of proving that unions of open sets are also open. The participants also mention the age of the thread and ask for permission to continue the discussion.
  • #1
Austin Chang
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  • #2
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.
 
  • #3
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 ?
 
  • #4
WWGD said:
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.
 
  • #5
PeroK said:
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.
 
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  • #6
@PeroK , I just spoke with Greg, he seems to be ok with it. You may chime in if you wish.
 

FAQ: Can someone explain these two examples of an open set

1. What is an open set in mathematics?

An open set in mathematics is a set of points that does not include its boundary points. In other words, for any point in the set, there exists a neighborhood around that point that is also included in the set. This concept is important in topology and analysis, and is used to define properties such as continuity and convergence.

2. How is an open set different from a closed set?

An open set does not include its boundary points, while a closed set includes all of its boundary points. This means that a closed set can be thought of as the complement of an open set. In other words, a set is closed if and only if its complement is open.

3. Can you provide an example of an open set?

One example of an open set is the set of all real numbers between 0 and 1, denoted as (0,1). This set does not include its boundary points 0 and 1, and for any point x in the set, there exists a neighborhood around x that is also included in the set.

4. What is the importance of open sets in mathematics?

Open sets are important in mathematics because they allow us to define and study important concepts such as continuity, convergence, and connectedness. They also provide a framework for defining more complex sets and topological spaces.

5. How are open sets used in real-world applications?

Open sets are used in real-world applications such as computer graphics, where they are used to define smooth surfaces and boundaries. They are also used in physics and engineering, where they are used to model and analyze systems that exhibit continuous behavior.

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