Difference between topology and sigma-algebra axioms.

Question

One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under countable union. It is very clear mathematically but is there a way to think; so that we can define a geometric difference? In other words I want to have an intuitive idea in application of this objects.

Answer 1

I would like to mention that in An Epsilon of Room, remark 1.1.3, Tao states:

The notion of a measurable space (X, S) (and of a measurable function) is superficially similar to that of a topological space (X, F) (and of a continuous function); the topology F contains ∅ and X just as the σ-algebra S does, but is now closed under arbitrary unions and finite intersections, rather than countable unions, countable intersections, and complements. The two categories are linked to each other by the Borel algebra construction.

Later, in example 1.1.5:

given any collection F of sets on X we can define the σ-algebra B [ F ] generated by F , defined to be the intersection of all the σ-algebras containing F , or equivalently the coarsest algebra for which all sets in F are measurable. (This intersection is non-vacuous, since it will always involve the discrete σ-algebra 2^X). In particular, the open sets F of a topological space ( X, F ) generate a σ-algebra, known as the Borel σ-algebra of that space.

Answer 2

Your question is a little vague, but here is something to consider: Topology is normally discussed as its own subject while $\sigma$-algebras are typically just used as a tool in measure theory. One reason why finite intersections are needed in a topology is that it preserves what we think of as "openness" in a metric space. For instance, the finite intersection of any intervals of the form $(a,b) \subseteq \mathbb{R}$ still has the property of containing a ball around each point. This property is not shared with $\sigma$-algebras. For instance we can consider the countable intersection

$$ \bigcap_{n \in \mathbb{N}} \left(a - \frac{1}{n}, b+ \frac{1}{n} \right ) \;\; =\;\; [a,b] $$

which doesn't preserve this "openness" property we would like a topology to preserve. We can see that every neighborhood around either points $a$ or $b$ contain elements outside the interval $[a,b]$.

Answer 3

An easy way to get a feeling for this is to consider basic examples.

For example, let $X=\{1, 2, 3\}$.

A topological space $(X, )$ could be for constructed by choosing for example $=\{∅,\{1, 2\},\{2\},\{2,3\},X\}$.

But this is as far from a $σ$-algebra as you can get since in fact no complement of any set in $$ is in except for $X$ and $∅$.

Have a look at some examples of topologies, some examples of $σ$-algebras and try to compare them. Start easy (like this) and move on to some harder ones and you will develop an intuition after hand.

Answer 4

The geometry of $\sigma$-algebras is in general badly understood. Proofs involving topologies often work directly on the topology, proving sets are open directly. The proof of Arzela-Ascoli, for example, works in two topologies and proves convergence directly. A great many proofs start with pick $U$ a neighborhood of $x$. Working with $\sigma$-algebras is somewhat more complicated. Often the approach is to build a sequence of approximations to the desired $\sigma$-algebra. Even the definition of the Borel algebra generated by some sets $\mathfrak{B}$ is either very abstract or build through approximations. I.E. the intersection of all $\sigma$-algebras containing $\mathfrak{B}$, or transfinite induction on stages of approximations to the full Borel algebra.

Answer 5

I just wanted to add a remarkable comment on this as I was reading through a great book on probability theory by Achim Klenke on p.8

Differences between topologies and sigma-algebra

In contrast with σ -algebras, topologies are closed under finite intersections only i.e. why the answer of the previous question applies as a case in point. Furthermore, the topologies are also closed under arbitrary unions while σ-algebras need not be closed under arbitrary unions, only under countable unions.

I hope this clarifies the matter

Answer 6

Topology is a construct which attempts to formalize the qualitative notion of "neighborhood-ness" or "closeness" of elements, a fundamentally binary concept in terms of whether a point is in a neighborhood or not. Sigma-algebra is a construct which attempts to formalize the quantitative notion of a "volume" or "weight" of sets of elements, which is inherently tied to a continuous value (often from the real number system).

One is binary - are x and y in the same neighborhood or not? The other is continuous and is strongly coupled with the nature of the real number system. Both of them show resilience and consistency under certain operations and fragility to others, in a way which actually makes intuitive sense:

1. In a sigma-algebra if a set has a volume so will its complement (the volume of everything minus the volume of the set). In a topology if an element is in an open set, then it is endowed with some neighborhood-ness to the other elements in the open set. This has no implication on the question if the elements in the complement share a neighborhood. Maybe they do, maybe they dont. The neighborhood-ness of the elements in set A carries not information about the neighberhoodness-ness of elements in A complement.
2. In a sigma-algebre the countable union of sets with a volume should have a volume too: we keep adding numbers which eventually might converge or diverge, all good. But an uncountable sum of volumes might not have a volume, as a matter of an undefined sum. For a topology, any union of neighborhoods is still a union of neigherbhoods. "Neighberhood" is a binary statement here- are they near or not? Volume is not binary, it's bounded to constraints of the real number system.
3. The same applies to intersections: in a sigma-algebra a countable intersection of sets with volume has a volume too (if something has a volume, certainly a part of it will have a volume as well, inductively so). In a topology the intersection of neighborhoods is fragile when we keep reducing it - eventually the elements of the intersection will lose their neighberhoods (leading to the canonical example of the countable intersection of (-1/n, 1/n)).

The axioms we have here for topology and sigma-algebra represent what are essentially reasonable abstractions of the notions of neighborhood-ness and volume-ness. We could have determined that a complement of an open set is open too - neighborhood ness of a,b,c -will- guarantee neighborhood-ness of the complement elements d,e, but we dont. I think it's important to understand the relative arbitariness of our assumptions here. We take them because they make sense for us and because they are "nice" - they allow us to develop a rich and useful theory.