Connections Between Kuratowski Partitions of Baire Spaces, Measurable Cardinals, and Precipitous Ideals

Abstract

In this paper, we investigate the existence and properties of Kuratowski partitions (K-partitions), i.e., partitions of Baire spaces such that all subfamilies of such partition sum to a set with the Baire property. We focus on the inherent symmetries in their structure and prove their connections to existence of measurable cardinals and precipitous ideals. Our results reveal that the existence of a K-partition in any Baire or compact space is symmetrically reflected in metrizable and completely metrizable spaces, respectively, and we explore how these symmetries extend to the realm of set-theoretic ideals and large cardinals. We also outline possible connections with real-measurable cardinals, extensions of Lebesgue measure on the closed interval, and density topologies.

1. Introduction

In this paper, we examine connections between Kuratowski partitions of Baire spaces and other foundational concepts, such as measurable cardinals and precipitous ideals. A Kuratowski partition, or K-partition for short, is a particularly well-behaved partition of a Baire space. However, such partitions need not exist in ZFC, i.e., Zermelo–Fraenkel set theory together with the axiom of choice, and their existence is not equiconsistent with ZFC. Because of this, providing an explicit example is not possible. Nevertheless, their deep connections with other areas of mathematics make them a worthwhile subject of study. The motivation for this work stems from a desire to understand the intricate interplay between topological partitions and set-theoretic properties, particularly those involving large cardinals.

Their genesis traces back to the paper [1] by Kuratowski where he posed a question about functions with the Baire property from a completely metrizable space to a metrizable space. He asked exactly when and under what conditions such functions have to be continuous apart from a meager set. From the outset, it was known that this holds when the range of the function is separable. It was shown in [2] that this also holds when the domain of the function has weight at most $\mathfrak{c}$. Among other things, this means that any real Baire-measurable function is continuous except on a meager set. This is strikingly similar to the fact that Lebesgue measurable functions are continuous except on a set of Lebesgue measure zero, as proved by Luzin [3]. The main question was when the assumption of separability could be dropped. As was shown in [4], this question can be rephrased using K-partitions; i.e., a function is reducible to a continuous function precisely when a K-partition of the domain does not exist. This makes K-partitions a valuable tool in the study of Luzin-type reduction theorems. One of the earliest approaches to the subject was by Kunugi [5], a contribution that has recently been reviewed in [6].

In [7], model theoretic connections between K-partitions and measurable cardinals as well as precipitous ideals have been proven. We strengthen those results, showing among other things that the existence of a K-partition of a Baire space, satisfying one additional condition, implies the existence of a measurable cardinal. For more on precipitous ideals see [8,9].

Although originally K-partitions were defined for the algebra of sets with the Baire property and the ideal of meager sets, they can be generalized to other $\sigma$-algebras of sets that have an associated $\sigma$-ideal, for example utilizing formalism of category bases presented in [10]. This approach has proven useful in [11], where nonexistence of Kuratowski partition was vital in proving Luzin-type reduction theorems for (s)-measurable and completely Ramsey functions. Very similar partition techniques have been also used in [12,13].

Recent developments in the theory of K-partitions have been discussed in [14]. In that work, K-partitions were used to generalize results by Louveau and Simpson [15]. Their original results concerned the restriction of completely Ramsey functions to a base clopen set in such a way that the range is separable. The generalization allows for a relaxation of the conditions imposed on the domain of the function in the theorem. In [16], the authors have found a connection between K-partitions and the Gitik–Shelah theorem [17], which states that forcing with a $\sigma$-complete ideal over a set cannot be isomorphic to Cohen, random, Hechler, or Sacks forcing. The authors simplified several proofs from that work, which originally relied on generic ultrapower techniques. Other recent results worth noting include [18], where similar techniques were used to construct examples of sets that are simultaneously non-measurable with respect to several $\sigma$-algebras, including (s)-property. In [19], the authors employed game-theoretic methods to identify non-measurable unions of certain subfamilies of ideals.

2. Definitions and Known Facts

Throughout the whole paper we assume that we are dealing with Hausdorff topological spaces exclusively, and the space denoted by X is assumed to be Baire. By a Baire space we mean a topological space in which any countable intersection of open dense sets is non-empty, i.e., a space that is non-meager in itself. Some authors refer to such spaces as locally Baire space.

A subset $A\subseteq X$ is said to have the Baire property if it can be represented as $U▵F$, where U is open, F is meager, and ▵ denotes the symmetric difference of sets. A partition $\mathcal{F}$ of X into meager sets is called a K-partition if for any ${\mathcal{F}}^{\prime }\subseteq \mathcal{F}$ the set $\bigcup {\mathcal{F}}^{\prime }$ has the Baire property.

Given a boolean algebra B and an ideal I on B, the saturation of I (denoted by $\mathrm{sat}\left(I\right)$) is the smallest cardinal such that all antichains in $B/I$ are of a cardinality less than $\mathrm{sat}\left(I\right)$. For an ideal I on a set Y (i.e., on an algebra $\mathcal{P}\left(Y\right)$), the set $W\subseteq \mathcal{P}\left(Y\right)$ is called an I-partition of Y if $\bigcup W=Y$, and for any $A_1,A_2\in W$, if $A_1\ne A_2$, then $A_1\cap A_2\in I$. The ideal I is called precipitous if it is $\left|Y\right|$-complete and for any sequence $W_0,W_1,\dots$ of I-partitions such that $W_{n+1}$ is a refinement of $W_n$ there exist $X_i\in W_i$ such that ${\displaystyle \bigcap _{i\in \omega }}{X}_i\ne \varnothing$.

For any ideal I on a cardinal $\kappa$ we will denote $I^{+}=P\left(\kappa \right)\setminus I$. Now let

$$X\left(I\right)=\{x\in {\left(I^{+}\right)}^{\omega }:{\forall }_{n\in \omega }\bigcap _{k=0}^{n}x\left(k\right)\in I^{+},\bigcap _{k<\omega }x\left(k\right)\ne \varnothing \}$$

be a subspace of the metric space ${\left(I^{+}\right)}^{\omega }$ where $I^{+}$ is a discrete space. It has been shown in [7] that I is precipitous if and only if $X\left(I\right)$ is a Baire space, and if I is precipitous then the sets

$$F_{\alpha }=\{x\in X\left(I\right):\alpha =min\bigcap _{k<\omega }x\left(k\right)\}\mathrm{for}\alpha \in \kappa$$

define a K-partition of $X\left(I\right)$.

Let ${\tau }^{+}$ be a discrete space of all non-empty open subsets of X. Similarly, we can define

$$X\left(\tau \right)=\{x\in {\left({\tau }^{+}\right)}^{\omega }:\bigcap _{k<\omega }x\left(k\right)\ne \varnothing \}$$

and

$$X^{*}\left(\tau \right)=\{x\in {\left({\tau }^{+}\right)}^{\omega }:\bigcap _{k<\omega }x\left(k\right)\ne \varnothing ,{\forall }_{n\in \omega }\mathrm{Cl}\left(x(n+1)\right)\subseteq x\left(n\right)\}.$$

As we will see those spaces will be vital in showing that if there exists a K-partition of a Baire space then there also exists a K-partition of some metric space. This in fact will give us the equiconsistency of existence of a measurable cardinal and the existence of a K-partition of any Baire space, as it was shown in [7].

 Theorem 1.  

The following theories are equiconsistent:

• ZFC + existence of a measurable cardinal;
• ZFC + existence of a K-partition of a Baire metric space;
• ZFC + existence of a K-partition of a complete metric space.

If $\mathcal{F}=\{F_{\alpha }:\alpha \in \kappa \}$ is a K-partition of Baire space X then the set

$$I_{\mathcal{F}}=\{A\in P\left(\kappa \right):\bigcup _{\alpha \in A}{F}_{\alpha }\mathrm{is} \mathrm{meager}\}$$

is an ideal on $\kappa$. If $U\subseteq X$ is open and non-meager, then

$${\mathcal{F}|}_U=\{F_{\alpha }\cap U:\alpha \in \kappa \}$$

is a K-partition of U. Moreover, if $U\subseteq V$ then $I_{{\mathcal{F}|}_V}\subseteq I_{{\mathcal{F}|}_U}$, and thus $X\left(I_{{\mathcal{F}|}_U}\right)\subseteq X\left(I_{{\mathcal{F}|}_V}\right)$.

There is yet another equivalent approach to precipitous ideals. Given an ideal I on Y a family F of functions on subsets of Y to ordinals is a functional if $W_F=\{\mathrm{dom}\left(\phi \right):\phi \in F\}$ is an I-partition of Y. Furthermore, for two functionals $F,G$ we define $F<G$ if

• $W_F$ refines $W_G$;
• For $f\in F,g\in G$ such that $\mathrm{dom}\left(f\right)\subseteq \mathrm{dom}\left(g\right)$, we have $f\left(\alpha \right)<g\left(\alpha \right)$ for $\alpha \in \mathrm{dom}\left(f\right)$.

In [8], a following characterization of precipitousness was shown.

Theorem 2.

An ideal I is precipitous if and only if for no $S\in I^{+}$ there exists a sequence of functionals $F_0>F_1>\dots$.

In [20], one can find the following result attributed to Banach, called the localization theorem.

Theorem 3.

Let X be a topological space and let the sets $U_i$ for $i\in \kappa$ be open meager subsets of X. Then ${\displaystyle \bigcup _{i\kappa }}{U}_i$ is also meager.

The localization theorem has one important consequence, which can be stated as follows.

Corollary 1.

Let X be a Hausdorff and Baire space. There exists open subset $U\subseteq X$ such that U has no non-empty meager open subset.

Proof. 

By the localization theorem, the set $W=\bigcup \{V\subseteq X:V \mathrm{is} \mathrm{open} \mathrm{and} \mathrm{meager}\}$ is open and meager. W cannot be dense, because X is a Baire space. Then $U=\mathrm{Int}(X\setminus W)$ is as required. □

A space with such properties may be called globally Baire. Those are exactly the spaces in which a countable intersection of open and dense subsets is dense.

3. Results

3.1. Basic Results About K-Partitions

We begin by proving that the existence of any K-partition whatsoever implies the existence of K-partition of some metrizable space.

Theorem 4.

Let X be a space with K-partition $\mathcal{F}$ of minimal cardinality κ and let τ be the topology of X. Then the space $X\left(\tau \right)$ admits a K-partition.

Proof. 

First, we show that $X\left(\tau \right)$ is a Baire space. Let $G_i\subseteq X\left(\tau \right)$ be open and dense for each $i\in \omega$. Define

$$G_i^{\#}=\bigcup \{U_0\cap \dots \cap U_n:n\in \omega ,{\forall }_{x\in X\left(\tau \right)}((x\left(0\right)=U_0\wedge \dots \wedge x\left(n\right)=U_n)\Rightarrow x\in G_i)\}.$$

The set $G_i^{\#}$ can be viewed as a reflection of $G_i$ in the underlying space X. Conceptually, it is similar to the interior of the set of all branches of a tree, for example, as in the Cantor set. Since $G_i^{\#}$ is a union of open sets, it is open in X. Next, we show that it is dense. Let $U\subseteq X$ be open and

$$G_U=\{x\in X\left(\tau \right):x\left(0\right)=U\}.$$

The set $G_U$ is open in $X\left(\tau \right)$, and by the density of $G_i$, we have that $G_U\cap G_i$ is non-empty and open. Therefore, there exist open sets $U_1,\dots ,U_n$ such that $U\cap U_1\cap \dots \cap U_n\subseteq G_i^{\#}$, and thus $U\cap G_i^{\#}\ne \varnothing$, which proves that $G_i^{\#}$ is dense. Since X is a Baire space, there exists $y\in {\displaystyle \bigcap _{i\in \omega }}{G}_i^{\#}$, i.e., $y\in U_{i,1}\cap \dots \cap U_{i,n_i}\subseteq G_i^{\#}$. It follows that $x=(U_{1,1},\dots ,U_{1,n_1},U_{2,1},\dots )\in {\displaystyle \bigcap _{i\in \omega }}{G}_i$, and thus $X\left(\tau \right)$ is a Baire space as required.

Let

$${\tilde{F}}_{\alpha }=\{x\in X\left(\tau \right):\alpha =min\{\beta <\kappa :\bigcap _{k<\omega }x\left(k\right)\cap F_{\beta }\ne \varnothing \}\}\mathrm{for}\alpha \in \kappa .$$

We will show that the sets ${\tilde{F}}_{\alpha }$ define a K-partition of $X\left(\tau \right)$. They are clearly disjoint, and their union is the entirety of $X\left(\tau \right)$.

Let $A\subseteq \kappa$ be such that $F_A={\displaystyle \bigcup _{\alpha \in A}}{F}_{\alpha }$ is meager, i.e., $F_A={\displaystyle \bigcup _{i\in \omega }}{M}_i$, where $M_i$ are nowhere dense in X. Let

$$G_k=\{x\in X\left(\tau \right):{\exists }_{i\in \omega }x\left(i\right)\subseteq X\setminus M_k\}.$$

The sets $G_k$ are unions of base open sets in $X\left(\tau \right)$ and are thus open. Let $U_0,\dots ,U_n\subseteq X$ be open sets such that $U=U_0\cap \dots \cap U_n\ne \varnothing$. Since $M_k$ is nowhere dense, we have $U\setminus \mathrm{Cl}\left(M_k\right)\ne \varnothing$. Consider $G=\{x\in X\left(\tau \right):x\left(0\right)=U_0,\dots ,x\left(n\right)=U_n\}$. Then $x=(U_0,\dots ,U_n,U,U,U,\dots )\in G\cap G_k$, which shows that $G_k$ is dense. Thus, ${\tilde{F}}_A\subseteq \{x\in X\left(\tau \right):\bigcap x\cap F_A\ne \varnothing \}\subseteq {\displaystyle \bigcap _{i\omega }}(X\left(\tau \right)\setminus G_i)$ is meager.

Now, let $A\subseteq \kappa$ be such that $F_A={\displaystyle \bigcup _{\alpha \in A}}{F}_{\alpha }$ is non-meager. We know that $F_A$ has the Baire property, i.e., $F_A=U▵{\displaystyle \bigcup _{i\in \omega }}{M}_i$, where $M_i$ are nowhere dense in X. Moreover, they can be assumed to be closed. Let once more

$$G_k=\{x\in X\left(\tau \right):{\exists }_{i\in \omega }x\left(i\right)\subseteq X\setminus M_k\}$$

and

$$G_U=\{x\in X\left(\tau \right):{\exists }_{i\in \omega }x\left(i\right)\subseteq U\}.$$

We already know that the sets $X\left(\tau \right)\setminus G_k$ are nowhere dense. Let $x\in G_U\setminus {\displaystyle \bigcup _{i\in \omega }}(X\left(\tau \right)\setminus G_k)$. For some $n\in \omega$ we have $x\left(n\right)\subseteq U$, so $\bigcap x\subseteq U$. On the other hand, for all $n,i\in \omega$, we have $x\left(n\right)\cap M_k=\varnothing$, i.e., $\bigcap x\cap M_k=\varnothing$. It follows that

$$\bigcap x\subseteq U\setminus \bigcup _{i\in \omega }{M}_i\subseteq U▵\bigcup _{i\in \omega }{M}_i\subseteq F_A,$$

which in turn shows that $x\in {\tilde{F}}_A$, and thus $G_U\setminus {\displaystyle \bigcup _{i\in \omega }}(X\left(\tau \right)\setminus G_k)\subseteq {\tilde{F}}_A$. □

This result can be further refined in the case when we have a K-partition of a compact space.

Theorem 5.

Let X be a compact space with K-partition $\mathcal{F}$ of minimal cardinality κ and let τ be the topology of X. Then the space $X^{*}\left(\tau \right)$ is a complete metric space and has a K-partition.

Proof. 

First, we show that $X^{*}\left(\tau \right)$ is complete. Let ${\left({\xi }_n\right)}_{n\in \omega }$ be a Cauchy sequence in $X^{*}\left(\tau \right)$. It is also a Cauchy sequence in the complete space ${\tau }^{\omega }$, so it converges to some $x\in {\tau }^{\omega }$. Because ${\tau }^{\omega }$ is a product of discrete spaces, there have to exist $N_n\in \omega$ such that for $k\ge N_n$ we have ${\xi }_k\left(n\right)=x\left(n\right)$. We need to show that $x\in X^{*}\left(\tau \right)$. We can assume $N_{n+1}>N_n$. Then,

$$\mathrm{Cl}\left(x\left(n+1\right)\right)=\mathrm{Cl}\left({\xi }_{N_{n+1}}(n+1)\right)\subseteq {\xi }_{N_{n+1}}\left(n\right)={\xi }_{N_n}\left(n\right)=x_n.$$

Thus, $x=\in X^{*}\left(\tau \right)$, as required.

We define a K-partition on $X^{*}\left(\tau \right)$ in the exact same way as we did for $X\left(\tau \right)$, as follows:

$${\tilde{F}}_{\alpha }=\{x\in X^{*}\left(\tau \right):\alpha =min\{\beta <\kappa :\bigcap _{k<\omega }x\left(k\right)\cap F_{\beta }\ne \varnothing \}\}\mathrm{for}\alpha \in \kappa .$$

For the remainder of the proof, the same reasoning as for $X\left(\tau \right)$ is valid for $X^{*}\left(\tau \right)$. □

The transfer of K-partitions between different classes of spaces can be viewed as a manifestation of a deeper symmetry in the theory, reflecting the robustness of the partition property under topological transformations.

3.2. K-Partitions, Precipitous Ideals, and Measurable Cardinals

As it turns out, the existence of K-partitions directly implies the existence of everywhere precipitous ideals.

Theorem 6.

Let X be a space with K-partition $\mathcal{F}$ of minimal cardinality κ. Then there exists an open non-meager subset U of X such that $I_{{\mathcal{F}|}_U}$ is everywhere precipitous.

Proof. 

According to the Banach Localization Theorem, there exists an open subset U such that it has no non-empty meager open subsets. Suppose O is an open and non-meager subset of U such that $I_{{\mathcal{F}|}_O}$ is not precipitous. Then there exists $S\in I_{{\mathcal{F}|}_O}^{+}$ and a descending chain of functionals

$${\mathsf{\Phi }}_0>{\mathsf{\Phi }}_1>\dots$$

(1)

on S. We have ${\displaystyle \bigcup _{\alpha \in S}}{F}_{\alpha }\cap O=V▵M$, where V is open and M is meager. Let $W_i=W_{{\mathsf{\Phi }}_i}=\{\mathrm{dom}\left(\phi \right):\phi \in {\mathsf{\Phi }}_i\}$ be corresponding $I_{{\mathcal{F}|}_O}$-partitions. Note that they are also $I_{{\mathcal{F}|}_V}$-partitions.

Let $Y\in W_i$. Then there exists ${\phi }_Y\in {\mathsf{\Phi }}_i$ such that $\mathrm{dom}\left({\phi }_Y\right)=Y$. We have ${\displaystyle \bigcup _{\alpha \in Y}}{F}_{\alpha }\cap V=V_Y▵M_Y$, where $V_Y$ is open and $M_Y$ is meager. Note that the sets $V_Y$ are pairwise disjoint. Indeed, if it was not the case, their intersection would be open and non-empty and thus non-meager by our assumption. Let $f_Y:V_Y\to \kappa$ be given by

$$f_Y\left(x\right)=\alpha \mathrm{for} x\in F_{\alpha }.$$

(2)

From the maximality of $W_i$, the sets $V_i={\displaystyle \bigcup _{Y\in W_i}}{V}_Y$ are open and dense in V, and therefore, by the Baire theorem ${\displaystyle \bigcap _{i\in \omega }}{V}_i\ne \varnothing$.

Let $f_i:V_i\to \kappa$ be given by $f_i={\displaystyle \bigcup _{Y\in W_i}}{f}_Y$. Then from the properties of functionals, we have that

$${\forall }_{i\in \omega }{\forall }_{x\in V_{i+1}}{f}_i\left(x\right)>f_{i+1}\left(x\right),$$

that is,

$${\forall }_{i\in \omega }{\forall }_{x\in V_{i+1}}{f}_i\left(x\right)\ni f_{i+1}\left(x\right).$$

Take $x\in {\displaystyle \bigcap _{i\in \omega }}{V}_i$. Then we have

$$f_0\left(x\right)\ni f_1\left(x\right)\ni \dots$$

which is a contradiction, as a strictly decreasing sequence of ordinals does not exist. □

By [7] we already know that the existence of K-partitions and the existence of measurable cardinals are equiconsistent. Moreover, under some minor additional assumptions the existence of a K-partition implies the existence of a measurable cardinal.

Proposition 1.

Let X be a space with K-partition $\mathcal{F}$. If $\mathrm{sat}\left(I_{\mathcal{F}}\right)$$<\omega$, then there exists an open non-meager set U of X such that $I_{{\mathcal{F}|}_U}$ is maximal.

Proof. 

If $\mathrm{sat}\left(I_{\mathcal{F}}\right)$$<\omega$, then the quotient algebra $\kappa /I_{\mathcal{F}}$ is finite, and thus it has atoms $\left[A_1\right],\dots ,\left[A_n\right]$. Let $B_i={\displaystyle \bigcup _{\alpha \in A_i}}{F}_{\alpha }$. By the K-partition property, $B_1=U▵M$, where U is open and M is meager. Consequently, $U\cap B_i$ is meager for $i>1$ and therefore $I_{{\mathcal{F}|}_U}$ is maximal. □

Theorem 7.

Let X be a space with K-partition $\mathcal{F}$ of minimal cardinality κ. Let $X\left(I_{\mathcal{F}}\right)$ be complete. Then κ is a measurable cardinal.

Proof. 

From minimality with respect to the condition above, we obtain that $\kappa$ is regular. We will show that there exists an open subset U of X such that $I_{{\mathcal{F}|}_U}$ is a maximal ideal.

Suppose that $\mathrm{sat}\left(I_{\mathcal{F}}\right)$$\ge {\omega }_1$. There exists

$$\mathcal{A}=\{A_n\in I_{\mathcal{F}}^{+}:n\in \omega \}$$

such that $A_n\cap A_m\in I_{\mathcal{F}}$ for every $n\ne m$. As $I_{\mathcal{F}}$ is ${\omega }_1$-additive, we can replace $A_n$ with $A_n\setminus {\displaystyle \bigcup _{m\ne n}}{A}_m$, obtaining $A_n\cap A_m=\varnothing$ for every $n\ne m$. Let $B_n={\displaystyle \bigcup _{k\ge n}}{A}_k$.

Now, let $x_m\left(n\right)=B_n$ for $n\le m$, and $x_m\left(n\right)=B_m$ otherwise. Of course all $x_m\in X\left(I_{\mathcal{F}}\right)$ and ${\left(x_m\right)}_{m\in \omega }$ is a Cauchy sequence, but its limit is not in $X\left(I_{\mathcal{F}}\right)$, contradicting its completeness.

Thus, $\mathrm{sat}\left(I_{\mathcal{F}}\right)$ is finite and by the second of the above propositions there exists an open set U in X such that $I_{{\mathcal{F}|}_U}$ is maximal. By the theorem above we may assume that it is also everywhere precipitous. From the precipitousness it is also $\kappa$-complete, and thus it makes $\kappa$ a measurable cardinal. □

4. Discussion

4.1. Possible Connections with Real-Measurable Cardinals

There are reasons to believe that K-partitions may also be connected with real-measurable cardinals as well. If there exists a real-measurable cardinal $\kappa \le \mathfrak{c}$, then we know from [21,22] that there exists a $\kappa$-additive measure $\mu$ defined on $\mathcal{P}\left(\mathbb{R}\right)$ that extends the regular Lebesgue measure. Let us restrict that measure to the compact interval $[0;1]$ and define a Boolean algebra

$$B\left(\mu \right)=\mathcal{P}\left([0;1]\right)/{\mathsf{\Delta }}_{\mu }$$

where $\mathsf{\Delta }=\{A\subseteq [0;1]:\mu (A)=0\}$. A similarity between $B\left(\mu \right)$ and random real forcing should be noted. We might now consider a Stone space $X\left(\mu \right)=St\left(B\right(\mu \left)\right)$ and introduce the partition of $X\left(\mu \right)$ in the following way. Let $F_x$ be the filter on $B\left(\mu \right)$ generated by the elements $[(x-{\displaystyle \frac{1}{n}};x+{\displaystyle \frac{1}{n}})\cap [0;1]]$ and let ${\tilde{F}}_x$ be the family of all ultrafilters extending $F_x$. It is fairly easy to see that these families are disjoint, closed $G_{\delta }$, nowhere dense, and cover all of $X\left(\mu \right)$. This connection suggests a deep interplay between real-valued measurable cardinals, measure algebras, and the theory of K-partitions, which merits further research. It is worth noting that the construction of the sets ${\tilde{F}}_x$ is very similar to a construction that shows the existence of topologies (called density topologies) on $[0;1]$ stronger than the euclidean topology and such that the sets with the Baire property and meager sets coincide exactly with the Lebesgue measurable sets and Lebesgue null sets, respectively; see [23,24].

4.2. Conclusions and Further Developments

While some foundational results on Kuratowski partitions and their connections to large cardinals are well-known, this paper provides new symmetry-based perspectives and strengthens previous theorems. The main results are not direct replications but build upon and extend the existing literature, offering new insights into the structure and transfer of K-partitions.

One open problem is whether the families constructed in the context of real-measurable cardinals truly form a K-partition, or at least whether it is equiconsistent with ZFC. Further investigation is needed to clarify this connection and its implications for measure theory and topology.

Another interesting problem is whether K-partitions can be used to obtain a general Luzin-type reduction theorem for topologizable category bases under certain additional assumptions, thereby generalizing and connecting the results in [10,11,25,26].