Is an Open Set Still Open After Removing Finite Points?

[*melinda*]

Use the definition of an open set to show that if a finite number of points are removed, the remaining set is still open.
Definition:
A set is open if every point of the set lies in an open interval entirely contained in the set.

I'm a bit lost, but I think that I somehow need to show that when a finite number of points are removed, the remaining points are still in the open interval. The problem is that I cannot figure out why removing points would change the location of other points. If I erase points from the open interval, of course the remaining ones would still be in the open interval... right?

The next part of the question reads:
If a countable number of points is removed from an open set the remaining set is not open. To show this, demonstrate a counterexample. Yikes!

I'm really lost on this one. I cannot even see why there would be a difference between removing a countable versus finite number of points.

help...

 

[StatusX]

Removing points from an open interval makes it so it's no longer an open interval.

First show that the set consisting of the entire space (eg, the entire real line if you are working with the real numbers) minus one point is open. (Hint: is a point open or closed?). Then, use the fact that a finite number of intersections of open sets is an open set. For the second part, think of a sequence of points that converges towards a limit.

 

[quicknote]

I can provide some clarification on this topic. Let's start with the definition of an open set. An open set is a set where every point in the set lies in an open interval that is entirely contained within the set. This means that for any point in the set, there exists a small enough open interval around that point that is also entirely contained within the set.

Now, let's consider the first part of the question - do open sets stay open when a finite number of points are removed? The answer is yes. This can be easily shown using the definition of an open set. Let's say we have an open set S, and we remove a finite number of points from it. This means we are left with a set S' that contains all the remaining points from S. Since S is open, we know that for every point in S, there exists an open interval around that point that is also contained within S. By removing a finite number of points, we are not changing the location of any other points in S. Therefore, the remaining set S' will still have the same open intervals around each point, making it an open set as well.

Now, let's move on to the second part of the question - if a countable number of points is removed from an open set, is the remaining set still open? The answer is no. This can also be demonstrated using a counterexample. Let's consider the set S = (0,1), which is an open set. Now, let's remove all the rational numbers from this set. This means we are removing a countable number of points, but we are left with the set (0,1) - Q, which is not an open set. This is because for any point in this set, there does not exist an open interval around it that is entirely contained within the set. This is because the set now contains irrational numbers, which are not in the form of a/b where a and b are integers. Therefore, the remaining set is not open.

In conclusion, open sets do stay open when a finite number of points are removed, but not when a countable number of points are removed. This is because removing a finite number of points does not change the location of other points in the set, while removing a countable number of points can change the structure of the set and make it no longer satisfy the definition of an open set. I hope this explanation helps