# Are all probability spaces topologies?

1. Aug 4, 2008

### BWV

Rudin's Real & Complex Analysis makes this statement:

"... there are situations, especially in probability theory, where measures occur naturally on spaces without topology, or on topological spaces that are not locally compact. An example is the ... Weiner Measure"

A discrete probability space would not be locally compact, so that part I get

But given this definition of a topology:

A topological space is a set X together with T, a collection of subsets of X, satisfying the following axioms:

The empty set and X are in T.
The union of any collection of sets in T is also in T.
The intersection of any finite collection of sets in T is also in T.

Given that a probability space meets all these criteria, I do not understand the point above about "spaces without topology"

2. Aug 4, 2008

### Focus

I agree with you, sigma-algebra satisfies all those axioms. I will take the book out next time I go on campus. Seems quite interesting.

3. Aug 4, 2008

### mathman

Probability spaces (like measure spaces) are closed under COUNTABLE unions and intersections. Topological spaces are defined by open sets, which are closed under ALL unions and FINITE intersections.