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.