Why are topology and sigma-algebras defined in a similar way?

[jordanl122]

Hi, I've been studying topology over the last semester and one thing that I was wondering about is why exactly is topology defined the way it is?
For a refresher:
given a set X we define a topology, T, to be a collection of subsets of X with the following 3 properties:
1) the null set and X are elements of T
2) the union of any elements of T is also in T (infinite)
3) the intersection of any of the elements of T is also in T (finite)

I was reading some measure theory and sigma-algebras are defined in a similar way, so I was wondering if someone could shed some light for me.

thanks,
Jordan

 

[LeonhardEuler]

As I understand it, (which I am only beginning to) topology is defined this way because it is an abstraction of studying open sets in a metric space. The sets that satisfy those requirements are defined to be the open sets. Defining them this way allows one to study topology on even sets without metrics. Those properties are satisfied by open sets in a metric space so it is a true generalization.

 

[matt grime]

Hi, I've been studying topology over the last semester and one thing that I was wondering about is why exactly is topology defined the way it is?

because that is the generalization of the metric structure on R that has worked at as the correct one in which to do analysis

I was reading some measure theory and sigma-algebras are defined in a similar way, so I was wondering if someone could shed some light for me.

Light on what exactly?

Sigma algebras differ in one significant way: they are closed under complements, thus effectively saying that you need to be closed under arbitrary intersection and union.