Examples of infinite/arbitrary unions of closed sets that remain closed.

[wisvuze]

Hello, I am trying to think of examples of these. At the moment, I can only think of ( on R ) closed intervals being the union of single-point sets ( infinitely many, the ones inside ).. et c. I also think the cantor set is an example of this.
Are there more "natural" examples? Thank you for sharing.

 

[wisvuze]

Ah , I can always take the intervals [ - n , n ] , to get the entire real line ( this is both open and closed ). I guess that is a natural example! Also, if I take interval subspaces of R, I can do the same thing with [a, n ] n to infinity.. et c

 

[micromass]

Hi wisvuze!

Can you tell me why you need this?? It makes it easier to give examples that you'll like.

There are many examples of this phenomenon. For example

$$\bigcup_{n=0}^{+\infty}{[2n,2n+1]}=[0,1]\cup [2,3]\cup [4,5]\cup ...$$

is also closed!

 

[wisvuze]

Thanks! I think I'm comfortable with different examples now. I have been personally "investigating" the "neighbourhood-finite" property. That is, a family of sets An on a topological space X is neighbourhood-finite if for any x in X, there is a neighbourhood V of x so that V intersects An at most a finitely many indices n.

 

[blue_raver22]

Hello there, it's great to see that you are exploring the concept of infinite/arbitrary unions of closed sets. I would like to share some of my insights on this topic.

Firstly, let's define what we mean by an infinite/arbitrary union of closed sets. An infinite union refers to the collection of all the elements that belong to any of the sets in the union, while an arbitrary union refers to the collection of elements that belong to at least one of the sets in the union. So, an infinite/arbitrary union of closed sets would be the collection of elements that belong to at least one closed set in the union.

Now, let's consider some examples of infinite/arbitrary unions of closed sets that remain closed. The first example you mentioned, which is the union of closed intervals on the real line, is a great example. This is because a closed interval, such as [0,1], is a closed set and the union of infinitely many closed intervals, such as [0,1], [1,2], [2,3], etc., is also a closed set.

Another example is the Cantor set, as you mentioned. The Cantor set is constructed by removing the middle third of a line segment, then removing the middle third of the remaining segments, and so on. This process can be continued infinitely, resulting in a set that is closed and has no isolated points. The union of all the removed segments, which is an infinite/arbitrary union, is also a closed set.

There are also more "natural" examples of infinite/arbitrary unions of closed sets. For instance, consider the set of all finite subsets of a given set. This set is closed under complementation and union, meaning that the union of any number of finite subsets is also a finite subset and therefore closed.

In conclusion, there are various examples of infinite/arbitrary unions of closed sets that remain closed. These examples can be found in different mathematical structures such as the real line, Cantor set, or even in finite sets. I hope this helps in your exploration of this topic. Keep up the scientific curiosity!