The diference between algebra, sigma algebra and topology

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  • #1
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how can I understand the difference between algebra, sigma algebra and topology
If I take the set A that contains a,b,c,d,e,f
the set C that contains A,phi,{a},{b,c,d,e,f}
then C is algebra on A
and C is sigma algebra on A
and (A,C) is topological space
is that true?
what is the difference ?
give me example like the obove to see the difference between them?
 
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  • #2
mathman
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You have mixed up two different aspects of set theory. For finite sets there is nothing to say. For infinite sets the definitions are as follows:

in all cases, the entire space and the empty set are included.

An algebra is a collection of subsets closed under finite unions and intersections.
A sigma algebra is a collection closed under countable unions and intersections.
In either case, complements are also included.

A topology starts with the idea that certain sets are open. The requirement is that the collection include all unions. However the open set collection needs to be closed only under intersections of a finite number of subsets.
 
  • #3
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You have mixed up two different aspects of set theory. For finite sets there is nothing to say. For infinite sets the definitions are as follows:

in all cases, the entire space and the empty set are included.

An algebra is a collection of subsets closed under finite unions and intersections.
A sigma algebra is a collection closed under countable unions and intersections.
In either case, complements are also included.

A topology starts with the idea that certain sets are open. The requirement is that the collection include all unions. However the open set collection needs to be closed only under intersections of a finite number of subsets.

That's a great succinct set of definitions. I could not find anything this clear on the web or in a textbook. If you don't mind, I'm saving the full post (2362521) in my files.
 
  • #4
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You have mixed up two different aspects of set theory. For finite sets there is nothing to say. For infinite sets the definitions are as follows:

in all cases, the entire space and the empty set are included.

An algebra is a collection of subsets closed under finite unions and intersections.
A sigma algebra is a collection closed under countable unions and intersections.
In either case, complements are also included.

A topology starts with the idea that certain sets are open. The requirement is that the collection include all unions. However the open set collection needs to be closed only under intersections of a finite number of subsets.

Hello. I am having a great deal of trouble understanding some of the theory on sigma algebra.

"An algebra is a collection of subsets closed under finite unions and intersections."
I'm not able to understand this one. For a sigma algebra, if the field contains, say, 20 elements, then the union of any two elements in this field must be in the field. But how is it any different for an algebra. Are the unions of only some of the elements in the field and not all of them? Does the difference come into play for only infinite sets?

In the definitions, for an algebra the unions go upto "n", for a sigma algebra the unions go upto infinity. Is "n" the number of elements in the algebra?

Also, can you recommend something to read that will help me understand the language used in the definitions? It took me a long time to be get an idea of what countable unions means and some thing that i can go to as a reference would be very helpful.
 
  • #5
811
6
"An algebra is a collection of subsets closed under finite unions and intersections."
I'm not able to understand this one. For a sigma algebra, if the field contains, say, 20 elements, then the union of any two elements in this field must be in the field. But how is it any different for an algebra. Are the unions of only some of the elements in the field and not all of them? Does the difference come into play for only infinite sets?


A finite sigma algebra is also an algebra. The key word in the definition of a sigma algebra is the word "countable". Any subset of a finite set is going to be countable.


In the definitions, for an algebra the unions go upto "n", for a sigma algebra the unions go upto infinity. Is "n" the number of elements in the algebra?

You don't need to concern yourself with what 'n' is. The key word here is "finite".


Also, can you recommend something to read that will help me understand the language used in the definitions? It took me a long time to be get an idea of what countable unions means and some thing that i can go to as a reference would be very helpful.

This is basic set theory stuff. Set theory is the basis of almost all modern mathematics, and it's a good investment to teach yourself all about it. The book I learned from was "Set theory and logic" By Robert Roth Stoll. It's not too shabby.
 

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