Whats an infinite intersection of open sets

In summary: I gather, it seems like the limit could be realized in different ways for an ∞ intersection of open sets... but as far as an example goes, i'm not sure if anyone has one.In summary, an infinite intersection of open sets is not necessarily open.
  • #1
natasha d
19
0
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?
 
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  • #2
natasha d said:
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 [itex]B_i[/itex] is the open set [itex](0, 1+ 1/n)[/itex] then the intersection of all of such [itex]B_i[/itex] is the half open interval [itex](0, 1][/itex].

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 [itex][1, 2][/itex].
 
  • #3
thanks. that really makes it so much clearer
 
  • #4
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.
 
  • #5
natasha d said:
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.
 
  • #6
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
 
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  • #7
Fredrik said:
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? :confused:

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 = [itex]\bigcup[/itex][itex]_{rεQ}[/itex]{(ry,y)/ y[itex]\in[/itex]{R}},

where {Q}, is the set of rational numbers, which is a countable set.
 
  • #8
natasha d said:
so there's no difference between an infinite intersection and a finite intersection? :confused:
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.
 
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  • #9
Fredrik said:
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.
 

1. What is an infinite intersection of open sets?

An infinite intersection of open sets is a mathematical concept where an infinite number of open sets are intersected to form a single set. An open set is a set that does not include its boundary points, and an infinite intersection means that the number of sets being intersected is infinite.

2. How is an infinite intersection of open sets different from a finite intersection?

An infinite intersection of open sets is different from a finite intersection in that a finite intersection involves only a finite number of sets being intersected, while an infinite intersection involves an infinite number of sets.

3. What is the significance of an infinite intersection of open sets?

An infinite intersection of open sets is significant in mathematics as it allows for the creation of more complex sets by intersecting an infinite number of simpler sets. It also has practical applications in topology and measure theory.

4. Can an infinite intersection of open sets be empty?

Yes, an infinite intersection of open sets can be empty. This can occur when the sets being intersected do not have any common elements. In this case, the resulting set will be empty.

5. How is an infinite intersection of open sets used in real-world applications?

An infinite intersection of open sets is used in real-world applications in fields such as computer science, physics, and economics. It is used to model complex systems and solve problems related to optimization, continuity, and convergence.

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