Topology on the set of partitions

Question

Let $X$ be the set of all partitions of $[0,1]$ such that each element of the partition is Lebesgue-measurable.

Let $Y$ be the set of all partitions of $[0,1]$ such that each element of the partition is a Borel set.

Is there a standard topology for a set like $X$ or $Y$? If so, is $X$ (or $Y$) compact in this topology?

I thought that the set of all partitions of $[0,1]$ (including those with Vitali sets as elements) is a more complicated object, but as @JDH points out in the proposed answer, this may not be the case.

The broader context for this question is my interest in convergence of measures on such a set (hence the focus on Borel or measurable partitions). If $Y$ is compact and metrizable, then the set of measures on $Y$ is itself a compact metric space in the weak topology, which opens the door to standard results on convergence of such measures.

Answer 1

There are several natural topologies to put on the space of partitions, which can also be thought of as the space of equivalence relations.

First, the collection of partitions of a set is a partial order under the refinement relation, where one partition $\mathcal{A}$ refines another $\mathcal{B}$ when every element of $\mathcal{A}$ is a subset of a set in $\mathcal{B}$. What's more, the space of partitions for each of your spaces is a lower semi-lattice, since any two partitions $\mathcal{A}$ and $\mathcal{B}$ have a coarsest common refinement, the collection of nonempty $A\cap B$, where $A\in\mathcal{A}$ and $B\in\mathcal{B}$. Indeed, the cases of $X$ and $Y$ lead to a $\sigma$-complete lower semi-lattice, since one may similarly intersect countably many sets and remain Borel or Lebesgue measurable as required. Since every partial order has a natural topology, the lower-cone topology, where the open sets are simply those that are closed downwards with respect to the order, we may place this topology on the space of partitions. In your case, the open sets of partitions would be those that include all refinements of any partition in the set.

This topology is compact---in the cases of $X$ and $Y$ and also in the case of the set of all partitions---simply because there is a coarsest partition, the partition with only one component consisting of the whole set. The only open neighborhood of this partition in the lower-cone topology is the entire space of partitions, since every partition refines it.

Secondly, a different dual topology would arise from turning the order upside down, and for that, the open sets would be closed under encoarsening rather than refinement. This topology also is compact, since there is a finest topology, consisting of the collection of singletons, and every partition is coarser than it.

Note that the space of all partitions, despite your remark on its complexity, actually exhibits much nicer lattice-theoretic properties than the Borel partitions or the measurable partitions. This is because the space of all partitions is actually a complete lattice. This is because any family of partitions has a largest common refinement, obtained simply by intersecting the equivalence relations as sets of ordered pairs. Similarly, one may take the equivalence relation generated by any family of partitions to find the smallest common encoarsening.

Another natural topology arises on the class of equivalence relations by saying that a basic open set is determined by finitely many equivalences and non-equivalences. Thus, one specifies finitely many equivalencies and non-equivalencies on the points $x_i\sim x_j$, $y_s\not\sim y_t$, and then the basic open set is the collection of all partitions that respect those finitely many requirements. This is a common kind of topology to place on the set of models of a first-order theory, where the basic open sets specify finitely much information about the predicates of the model; the collection of partitions amounts to the collection of equivalence relations, which is first-order.

One can imagine more elaborate topologies in this line, which take more into account your context of Lebesgue measurable or Borel partitions, by moving beyond finitely many point requirements to the case of finitely many requirements on Borel sets, or on measurable sets.

It would of course depend on your purpose with the topology to know which is best for that purpose.

Finally, let me mention that the well-developed theory of Borel equivalence relations is deeply concerned with the space of Borel partitions of the reals, but not exactly in your sense. What is usually required in that theory is not merely that every element of the partition is Borel, that is, that every equivalence class is Borel, but rather one insists that the binary relation itself, thought of as a subset of the plane, should be Borel. This is a more uniform kind of Borel, and the theory is extremely robust and active.

Answer 2

I'd like to mention an alternative to Prof. Hamkins' answer. His answer seems to be natural from the point of view of lattices; what I mention below can be interpreted as natural from the point of view of measure/information/entropy/ergodic theory, and it goes back to Rohlin (at least in the way I present it here).

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(I won't get into detail after this section.)

Let $(M,\mathcal{B},\mu)$ be a probability space that is measure theoretically isomorphic to the closed unit interval with Lebesgue $\sigma$-algebra and Lebesgue measure. (Such spaces are called Lebesgue or Lebesgue-Rohlin or standard or standard Borel.)

By a $\mu$-ae partition of $(M,\mathcal{B},\mu)$ I mean an $\alpha\subseteq \mathcal{B}$ such that

1. $\forall A_1,A_2\in\alpha: A_1\neq A_2\implies\mu(A_1\cap A_2)=0 $,
2. $\mu\left(M\setminus \bigcup_{A\in \alpha}A\right)=0$.

(So a $\mu$-ae partition is a collection of measurable subsets which partitions $M$ up to $\mu$-negligible sets.)

Denote by $\text{aefPar}(M,\mathcal{B},\mu)$ be the collection of all $\mu$-ae partitions of $(M,\mathcal{B},\mu)$, and by $\text{aefPar}(M,\mathcal{B},\mu), \text{aecPar}(M,\mathcal{B},\mu)$ and $\text{aemPar}(M,\mathcal{B},\mu)$ be the subcollections of finite, countable and measurable $\mu$-ae partitions of $(M,\mathcal{B},\mu)$, respectively. For the last collection defined, the adjective "measurable" signifies a certain separation property (and not just that each element of the partition is a measurable set): A partition $\alpha$ is measurable if the natural projection $M\to M/\alpha$ admits a Fubini Theorem.

Note that we are doing everything $\mu$-ae and $\mu$ is fixed, so we may safely consider only Borel subsets of $M$.

We can consider the existence of (ae-)refinements of these collections as an order relation as in Prof. Hamkins' answer.

I should note that the gnarly notation above is due to me; Rohlin himself uses $Z_1$ for $\text{aefPar}(M,\mathcal{B},\mu)$ and does not use any notation for the other collections, and nowadays typically no notation for any of these is in common use as far as I know.

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To any measurable ae-partition one can associate a (possibly infinite) nonnegative number called entropy, and similarly there is a notion of conditional entropy for any pairs of measurable ae-partitions; it is traditionally denoted by

$$H(\bullet\vert\bullet)=H_\mu(\bullet\vert\bullet):\text{aemPar}(M,\mathcal{B},\mu)\times \text{aemPar}(M,\mathcal{B},\mu)\to [0,\infty],$$

and $H=H_\mu= H_\mu(\bullet\vert \{M\})$ recovers the unary entropy, $\{M\}$ being the indiscrete partition of $M$.

(See Why is "h" used for entropy? for the claim that this letter is capital eta.)

I will not give the definition of these numbers here; see e.g. https://www.merry.io/dynamical-systems/26-partitions-and-the-rokhlin-metric/ for an account that seems to be very closely following Katok & Hasselblatt's Introduction to the Modern Theory of Dynamical Systems, pp.161-167 (though what they call a "measurable partition" there is what I call a finite or countable ae-partition here; my terminology is in sync with Rohlin's and also coincidentally the current common nomenclature in smooth ergodic theory as far as I know).

Conditional entropy has many nice properties compatible with the order relation we mentioned above. As its name would suggest, it is not symmetric; symmetrizing it (e.g. by considering $d_\mu:(\alpha,\beta)\mapsto H_\mu(\alpha\vert\beta)+H_\mu(\beta\vert\alpha)$) gives a (possibly infinite) distance function on $\text{aemPar}(M,\mathcal{B},\mu)$:

$$d_\mu:\text{aemPar}(M,\mathcal{B},\mu)\times \text{aemPar}(M,\mathcal{B},\mu)\to [0,\infty].$$

The common practice is to start with the smaller and more well-behaved subcollections and then verify that the definition admits extensions to the larger collections (One can even go beyond measurable ae-partitions; it is not reasonable to expect the ae-partitions coming from dynamical systems to be measurable. I won't get into these matters here.). Following Rohlin, let us focus on the subspace $\{H_\mu<\infty\}$ of measurable ae-partitions with finite entropy (Rohlin denotes this subspace by $Z$).

Theorem (Rohlin):

1. $(\{H_\mu<\infty\},d_\mu)$ is a complete separable metric space.
2. $\text{aefPar}(M,\mathcal{B},\mu)$ is dense in $(\{H_\mu<\infty\},d_\mu)$.
3. $\text{aemPar}(M,\mathcal{B},\mu)$ has a Polish space structure compatible with $d_\mu$ on $\{H_\mu<\infty\}$ is dense. W/r/t this structure $\{H_\mu<\infty\}$ is dense in $\text{aemPar}(M,\mathcal{B},\mu)$.
4. Both the unary and the binary $H_\mu$ are continuous w/r/t these structures (to say the least).

I should mention that in practice the finiteness of entropy is often automatic so infinite entropy is often not a big problem.

For more details and proofs of the above statements see Rohlin's paper "Lectures On The Entropy Theory Of Measure-Preserving Transformations".

Finally in the Katok-Hasselblatt book and the notes I've given a link to above another distance function is defined on $\text{aefPar}(M,\mathcal{B},\mu)$, which is essentially an extrapolation of the idea of metrizing the measure algebra by the measure of the symmetric difference. This measure distance is compatible in a certain sense with the entropy distance as is noted in the notes and the Katok-Hasselblatt book. Katok-Hasselblatt calls the entropy distance the Rohlin distance.

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I claimed in the beginning that the topology I'll be describing will be natural. This claim is based on the naturality of the notion of conditional entropy as used here: it is the unique (up to a normalization) function that satisfies certain properties one would expect from a quantification of surprise. (The notes I've cited above attributes the standard theorem along these lines to Khinchine; see also the exercise on p. 11 of https://math.huji.ac.il/~mhochman/preprints/info-theory.pdf)