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Whats an infinite intersection of open sets 
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#1
Mar712, 06:18 AM

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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 (11/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? 


#2
Mar712, 09:25 AM

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PF Gold
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#3
Mar712, 10:00 AM

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thanks. that really makes it so much clearer



#4
Mar712, 11:18 AM

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Whats an infinite intersection of open sets
If ##\{E_ii\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. 


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Mar712, 01:18 PM

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#6
Mar912, 07:57 PM

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a countable intersection of open sets is called a G delta set, and a countable union of closed sets is called an Fsigma 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 Gdelta set. you can google those terms for more.
http://en.wikipedia.org/wiki/Gδ_set 


#7
Mar1612, 12:01 PM

P: 19

does anyone have an example of an ∞ intersection of open sets thats open? Also does an ∞ intersection always have to be nested? thanks for the link mathwonk (couldnt fully fathom it) This is the first time ive seen a union of lines treated as a union of sets. a very interesting approach to obtaining a line of rationals only 


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Mar1612, 12:13 PM

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#9
Mar1612, 12:38 PM

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