Topologies vs Measure Theory’s Sigma-Algebras

Topologies and measure theory’s sigma-algebras look superficially similar, but the differences in their defining axioms lead to differences in their resulting study. In this post, we write this out to understand this better.

We lay the similarities and differences out in a chart to better clarify this situation:

Contains:	Topology	Sigma-Algebra
Empty set and whole set	Yes	Yes
Finite union	Yes	Yes
Countable union	Yes	Yes
Arbitrary union	Yes	No
Finite intersection	Yes	Yes
Countable intersection	No	Yes
Arbitrary intersection	No	No
Complement	No	Yes

As we can see, the differences are in arbitrary union (topology), countable intersection (sigma-algebra), and complement (sigma-algebra.) Thus, consider a structure that is defined by containing:

1. Empty set and whole set
2. Finite union
3. Countable union
4. Finite intersection

Or, more concisely:

1. Empty set and whole set
2. Countable union
3. Finite intersection

This is exactly the “middle ground” or “intersection” between topologies and sigma-algebras. In fact, this is true in a precise sense: the intersection of the classes of topologies and sigma-algebras is exactly the class of these structures.

As this answer points out, the “Epsilon of Room” document by Terence Tao delves further into the relationship between these two concepts, mentioning the Borel algebra construction linking them. I would need to learn more to understand those connections.

Also, is there a standard name for this “common ground” structure? The concept of a -system covers only finite intersections, while the concept of a -system requires complement and disjoint unions. However, there may be some logical implications that allow a related concept to work, since as stated in this Wikipedia article any family that is both a -system and a -system is also a sigma-algebra, which seems much richer at first glance.