Whats an infinite intersection of open sets

[natasha d]

whats an infinite intersection of open sets? how is it different from finite intersection of open sets
and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit?

it really does look look like a limit in the case of ∞ intersections, as in the sets are tending towards their intersection but not actually attaining it . Consider the intersection of the sets
∞
π (1-1/n, 2+ 1/n)
n=1
would the smallest set be an infinitesimally small ε on either side of the closed set [1,2], which would hence be their infinite intersection?

 

[HallsofIvy]

whats an infinite intersection of open sets? how is it different from finite intersection of open sets

Uhh... it involves an infinite collection of open sets rather than finite?

and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit?

First, yes, anything involving "infinity" has to be a limit because "infinity" is not a real number. However, it is NOT true that the intersection of an infinite number of open sets must be closed. For example, if B_i is the open set (0, 1+ 1/n) then the intersection of all of such B_i is the half open interval (0, 1].

It really does look look like a limit in the case of ∞ intersections, as in the sets are tending towards their intersection but not actually attaining it . Consider the intersection of the sets
∞
π (1-1/n, 2+ 1/n)
n=1
would the smallest set be an infinitesimally small ε on either side of the closed set [1,2], which would hence be their infinite intersection?

It's really not a good idea to talk about "infinitesmally small" numbers- that involves really, really deep and complicated concepts I prefer to avoid! Rather, I would say that if x is any number less than 1, there exist n such that x< 1- 1/n and so x is not in the interval for that n and so not in the intersection. If x> 2, there exist n such that 2+ 1/n< x and so that x is not in the intersection. Obviously, 1, 2, and all numbers between them are all in every such interval and so in the intersection- that intersection is [1, 2].

 

[natasha d]

thanks. that really makes it so much clearer

 

[Fredrik]

If $\{E_i|i\in I\}$ is a collection of sets, then $\bigcap_{i\in I}E_i$ is said to be a finite intersection if I is finite, and an infinite intersection if I is infinite.

I wouldn't say that an infinite intersection is defined as a limit. Maybe it can be, but that's not usually how it's done. $\bigcap_{i\in I}E_i$ is the set of all x such that $x\in E_i$ for all $i\in I$. This is true regardless of whether $I$ is finite, countable, or uncountable.

 

[SteveL27]

whats an infinite intersection of open sets? how is it different from finite intersection of open sets
and why is it a closed set in the case of ∞ intersection but open in case of finite.

An infinite intersection of open sets is not necessarily open. Several people lately have asked the same question, thinking that an infinite intersection of open sets is closed. It MIGHT be, but it might not be. All we know for sure is that an infinite intersection of open sets need not be open.

 

[mathwonk]

a countable intersection of open sets is called a G -delta set, and a countable union of closed sets is called an F-sigma set. these are rather interesting as not all subsets can occur this way. E.g. any countable set such as the rationals is F sigma, but i believe the set of rationals is not a G-delta set. you can google those terms for more.

http://en.wikipedia.org/wiki/Gδ_set

 

[natasha d]

If $\{E_i|i\in I\}$ is a collection of sets, then $\bigcap_{i\in I}E_i$ is said to be a finite intersection if I is finite, and an infinite intersection if I is infinite.

I wouldn't say that an infinite intersection is defined as a limit. Maybe it can be, but that's not usually how it's done. $\bigcap_{i\in I}E_i$ is the set of all x such that $x\in E_i$ for all $i\in I$. This is true regardless of whether $I$ is finite, countable, or uncountable.

so there's no difference between an infinite intersection and a finite intersection?

All we know for sure is that an infinite intersection of open sets need not be open.

i guess that's because of the way the infinite sets are defined i.e. with respect to an n.
does anyone have an example of an ∞ intersection of open sets that's open?
Also does an ∞ intersection always have to be nested?

thanks for the link mathwonk (couldnt fully fathom it) This is the first time I've seen a union of lines treated as a union of sets. a very interesting approach to obtaining a line of rationals only

For example, the set A of all points (x,y) in the Cartesian plane such that x/y is rational is an Fσ set because it can be expressed as the union of all the lines passing through the origin with rational slope:

A = \bigcup_{rεQ}{(ry,y)/ y\in{R}},

where {Q}, is the set of rational numbers, which is a countable set.

 

[Fredrik]

so there's no difference between an infinite intersection and a finite intersection?

There is. The number of sets that are being intersected. But it's not a different operation on the sets. $\bigcap_{i\in I}E_i$ is always the set of all x such that $x\in E_i$ for all $i\in I$, regardless of whether I is finite, infinite but countable, or uncountable.

 

[natasha d]

There is. The number of sets that are being intersected. But it's not a different operation on the sets. $\bigcup_{i\in I}E_i$ is always the set of all x such that $x\in E_i$ for all $i\in I$, regardless of whether I is finite, infinite but countable, or uncountable.

i meant the end result, the final set that we look at.