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The axioms of $\sigma$-algebra and of open sets seem quite similar to me.

Topology has subsets closed under finite intersection and arbitrary union. $\sigma$-algebra has subsets closed under complementation and under countable union (implying it is also closed under countable intersections). The definition of a $\sigma$-algebra also implies that it also includes the empty subset. So both definitions include empty set and the whole set.

My question is: does it even make sense to try to relate these concepts? I know $\sigma$-algebra from measure theory, while in topology, we do not measure things, but perhaps measurable spaces and topological spaces could share some interesting properties?

Remark: I am using the definition of topology via open sets.

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    $\begingroup$ The Borel $\sigma$-algebra connects the two. There are properties like $\mathcal B(\mathbb R\times\mathbb R)=\mathcal B(\mathbb R)\times\mathcal B(\mathbb R)$. $\endgroup$
    – nejimban
    Commented Sep 5, 2021 at 17:18
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    $\begingroup$ There are also certain axiom schema where the concepts coincide; as I understand it, the difference mostly involves very noisy "choice-y" sets. $\endgroup$ Commented Sep 5, 2021 at 17:20
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    $\begingroup$ A countable discrete topology is also a $\sigma$ -algebra. The class of all subsets (open sets)is the class of all measurable sets. $\endgroup$
    – ali reza
    Commented Sep 5, 2021 at 17:34
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    $\begingroup$ This question might help you: math.stackexchange.com/questions/3793828/… $\endgroup$ Commented Sep 5, 2021 at 17:53
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    $\begingroup$ @nejimban (and others interested): Regarding $\mathcal B(\mathbb R\times\mathbb R)=\mathcal B(\mathbb R)\times\mathcal B(\mathbb R)$ and related issues for Lebesgue measurable sets, see my 5 May 2002 sci.math.research post and other posts in that same thread. $\endgroup$ Commented Sep 5, 2021 at 18:20

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You're not wrong in thinking that the definitions of "topological space" and "measurable space" are similar! You're also not wrong in thinking that these similar axiomatizations might lead to "similar looking" structures, and the question is "can we make this precise?". The answer is, of course, "yes". Analyzing the similarities between constructions is one of the things Category Theory excels at, and indeed we can apply it here. In a way that we're about to make precise, the category of topological spaces (with continuous maps) $\mathsf{Top}$ and the category of measurable spaces (with measurable maps) $\mathsf{Meas}$ are very similar.

The key notion is that of a topologically concrete category. This is a category $\mathcal{T}$ equipped with a faithful functor $U : \mathcal{T} \to \mathsf{Set}$ that has certain nice properties inspired by the nice properties of the "Underlying Set" functor $U : \mathsf{Top} \to \mathsf{Set}$. You can read the all about them at the linked article, or in Adamek, Herrlich, and Strecker's The Joy of Cats, but I'll summarize a few important properties of topological categories here:

  1. They're complete and cocomplete. Moreover, we can construct limits/colimits in $\mathcal{T}$ by looking at the limit/colimit of underlying sets, then choosing "the right topology" (resp. $\sigma$-algebra, etc).

  2. It's possible that multiple spaces in $\mathcal{T}$ will have the same underlying set. We end up with a (possibly large) complete lattice of spaces over any set $X$. That is, given any family of topologies (resp. $\sigma$-algebras, etc) on $X$, there is a unique largest topology (resp. $\sigma$-algebra, etc) contained in each member of that family, and a unique smallest topology (resp. $\sigma$-algebra, etc) containing each member of that family.

  3. As a particular example of (2), every set $X$ has a discrete and indiscrete topology (resp. $\sigma$-algebra, etc). Moreover, every map from a discrete space is continuous (resp. measurable, etc) and every map to an indiscrete space is continuous (resp. measurable, etc). We can phrase this succinctly by saying that $U$ has left and right adjoints $\Delta \dashv U \dashv \nabla$ (sending a set to its discrete or indiscrete space respectively).

  4. Every morphism in $\mathcal{T}$ factors as a surjection (epi) followed by a subspace inclusion (regular mono).

There's much more to say about the nice properties enjoyed by topologically concrete categories, but suffice to say this framework provides a robust way of comparing $\mathsf{Top}$ with $\mathsf{Meas}$ -- categorically they are extremely similar!


I hope this helps ^_^

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