Difference between sigma algebra and topological space

  • #1
What is the difference between sigma algebra and topological space topological space?also what is the meaning of algebra on a set? the definitions are very similar except that in the case of sigma algebra the union is taken to include infinite number of sets .right?
 
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  • #2
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The difference between an algebra and a sigma-algebra is that we allow countable unions and intersections in sigma-algebra's. With algebra's, we only allow finite unions and intersections.

The difference between a topology and a sigma-algebra is this:
  • In sigma-algebra's we only allow countable unions, but in topologies, we allow arbitrary unions.
  • In sigma-algebra's we allow countable intersections, but in topologies we only allow finite unions.
  • In sigma-algebra's we allow complements, but in topologies we don't.
 
  • #3
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Topology: [itex](X,\tau)[/itex].

[tex]\text{T1.}\enspace\enspace(\emptyset\in\tau)\&(X \in\tau);[/tex]

[tex]\text{T2.}\enspace\enspace ((\forall i \in I)[A_i \in\tau ]) \Rightarrow \left ( \bigcup_{i \in I} A_i \in \tau \right ), \enspace I \text{ any index set};[/tex]

[tex]\text{T3.}\enspace\enspace (A_1,A_2,...A_n\in\tau)\Rightarrow \left ( \bigcap_{i =1}^n A_i \in \Sigma \right ).[/tex]

That is: T1. The empty set and (its complement), X, are in tau; T2. Each union of elements of tau is in tau (tau is "closed under unions"); T3. Each intersection of finitely many elements of tau is in tau (tau is "closed under finite intersections").

Note: This sense of the word "closed", applied to tau itself, is totally unrelated to the sense in "a closed set" (=the complement of an element of tau), applied to subsets of X.



Sigma Algebra: [itex](X,\Sigma)[/itex].

[tex]\text{S1.}\enspace\enspace\Sigma\neq\emptyset ;[/tex]

[tex]\text{S2.}\enspace\enspace(A \in\Sigma)\Rightarrow(X\setminus A \in\Sigma);[/tex]

[tex]\text{S3.}\enspace\enspace((\forall i \in \mathbb{N})[A_i \in\Sigma ]) \Rightarrow \left ( \bigcup_{i \in \mathbb{N}} A_i \in \Sigma \right ).[/tex]

That is: S1. Sigma is nonempty; S2. If a subset of X is in sigma, its complement is in sigma (sigma is "closed under complementats"); S3. Each union of countably many elements of sigma is in sigma (sigma is "closed under countable unions").

Equivalent forms of axiom S1, which make the resemblance to a topology seem even closer, are

[tex]\text{S1b.}\enspace\enspace\emptyset\in\Sigma;[/tex]

[tex]\text{S1c.}\enspace\enspace X \in\Sigma.[/tex]

An equivalent axiom to S3 is to require sigma to be closed under countable intersections. And yes, for an algebra of sets, replace S3 with the requirement that each union of finitely many elements of sigma is in sigma.
 

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