Topology and sigma-algebra

[Meital]

Can someone give me an example of a topology, which is not sigma algebra?

[matt grime]

A sigma algebra is closed under the operation of complements. Give me a topology (ie a collection of open sets) that is not closed under complements (pretty much most of the ones you can think of, I imagine)

[Meital]

I see, so we can take the colletion: R ( real numbers), open set (0,2), and the empty set. This is topology since it satisfies all 3 axioms of topology, but not sigma-algebra because we don't have the comp of (0,2) in the collection.

[matt grime]

well, that is an odd example to me, but works. R with its usual metric topology was the obvious example.

[Haroon Stephen]

On 9/6/2005 at 2:26 AM, Meital said:

Can someone give me an example of a topology, which is not sigma algebra?

For a set S={1,2,3} consider T={{ },  {1,2},  {2},  {2,3},  S}. T is a topology but not a sigma-algebra.