A Kuratowski closure-complement variant whose solution is independent of ZF
Michael P. Cohen, Todd Johnson, Adam Kral, Aaron Li, Justin Soll
aa r X i v : . [ m a t h . GN ] M a y A KURATOWSKI CLOSURE-COMPLEMENT VARIANT WHOSE SOLUTIONIS INDEPENDENT OF ZF
MICHAEL P. COHEN, TODD JOHNSON, ADAM KRAL, AARON LI, AND JUSTIN SOLL
Abstract.
We pose the following new variant of the Kuratowski closure-complement problem:How many distinct sets may be obtained by starting with a set A of a Polish space X , andapplying only closure, complementation, and the d operator, as often as desired, in any order?The set operator d was studied by Kuratowski in his foundational text Topology: Volume I ; itassigns to A the collection dA of all points of second category for A . We show that in ZFC settheory, the answer to this variant problem is 22. In a distinct system equiconsistent with ZFC,namely ZF+DC+PB, the answer is only 18. Introduction
Kuratowski’s closure-complement theorem , a result of his 1922 thesis, states that at most 14distinct sets are obtainable by applying the operations of closure and complementation to anyparticular initial set A in any topological space X . The algebraic result underlying this theoremis that the monoid generated by the set operators k (closure) and c (complement) has cardinality ≤
14. This surprising and amusing result has inspired a substantial literature of generalizationsand variants; see for example [4], [5], [11], [1], [3] or visit Bowron’s website
Kuratowski’s Closure-Complement Cornucopia [2] for a comprehensive list of relevant literature.The purpose of this note is to give an example of a natural variant of the Kuratowski closure-complement problem, whose solution turns out to be independent of ZF set theory. To pose theproblem, we let X be a topological space. Given a subset A ⊆ X , we say that a point p ∈ X isa point of second category for A if whenever U ⊆ X is an open neighborhood of p , we have U ∩ A nonmeager in X . Then, we define dA = { p ∈ X : p is a point of second category for A } .The operator d was apparently first defined by Kuratowski himself in his foundational text Topol-ogy Vol. 1 ([7], first edition 1933) and is associated with the application of Baire category methods ingeneral topology. The operator has handy applications in important descriptive set theoretic results,especially as it appears in Pettis’s lemma which states that if
A, B ⊆ X have the Baire property (Definition 1), then AB ⊇ id ( A ) id ( B ) (where i denotes the topological interior operator), and thus i ( AB ) is nonempty [8]. This lemma implies that every Borel-measurable homomorphism betweenPolish (i.e., separable and completely metrizable) topological groups is automatically continuous(see [9] for an admirable survey of this and many related results). We ask the following. Question.
How many distinct sets may be obtained by starting with a subset A of a Polish space X , and applying only the operators k , c , and d , as often as desired, in any order? Equivalently,what is the maximal cardinality of the monoid of set operators generated by k , c , and d ?For the remainder of this paper, X denotes an arbitrary Polish (separable completely metrizable)space, k and i the closure and interior operators on X respectively, and c the complementation Mathematics Subject Classification. operator. We recall the DeMorgan’s law for interiors and closures which states that kc = ci , orequivalently that ic = ck . We let KD denote the monoid of set operators on X generated by k , c , and d . We first answer the question in the traditional domain, where we assume the usual ZFaxioms plus the Axiom of Choice (AC). Theorem 1 (ZFC) . The cardinality of KD is ≤ . Moreover, if X = R with the usual topology,then there exists a set A ⊆ R for which { oA : o ∈ KD} = 22 . On the other hand, weak forms of AC are not sufficient to obtain the solution above. We denoteby DC the Axiom of Dependent Choice, which is equivalent over ZF to the Baire Category Theorem.We denote by PB the axiom that “every subset of every Polish space has the Baire property,” andwe recall the definition of the Baire property below.
Definition 1.
A set A ⊆ X has the Baire property if there exists an open set U ⊆ X for which thesymmetric difference A ∆ U = ( A − U ) ∪ ( U − A ) is a meager set.In the seminal paper [12], Solovay showed that if ZF is consistent with the existence of an inac-cessible cardinal, then ZF+DC+PB is consistent. In [10], Shelah improved this result to show thatZFC and ZF+DC+PB are equiconsistent axiom systems. Our second theorem below shows thatthe solution to this natural extension of the Kuratowski problem differs in this alternative axiomsystem, and thus the solution is independent of ZF. Theorem 2 (ZF+DC+PB) . The cardinality of KD is ≤ . Moreover, if X = R with the usualtopology, then there exists a set A ⊆ R for which { oA : o ∈ KD} = 18 . Preliminaries and Sets with the Baire Property
First we establish the basic properties of the operator d , most of which are observed withoutproof in [7] 4.IV. Lemma 3 (ZF+DC) . Let X be a Polish space, and A, B ⊆ X .(a) A ⊆ B implies dA ⊆ dB .(b) dA is closed and therefore dA ⊆ kA .(c) A open implies dA = kA .(d) d ( A ∪ B ) = dA ∪ dB .(e) A − dA is meager.(f ) A is meager in X if and only if dA = ∅ .(g) ddA = dA .(h) dkA = kikA .(i) kidA = dA .Proof. (a) Immediate from the definition of d . (b) Suppose p ∈ X is a limit point of dA . Given an arbitrary open neighborhood U of p , it meansthere is x ∈ dA with x ∈ U . Since x is a point of second category for A , U ∩ A is nonmeager in X ,whence p ∈ dA . (c) We have dA ⊆ kA by (c). Conversely if p ∈ kA , then each open neighborhood U of p willsatisfy U ∩ A = ∅ . Assuming A is open, then U ∩ A is a nonempty open set and hence nonmeagerby the Baire Category Theorem (equivalent to DC). Thus p ∈ dA . (d) By (a), we have dA ∪ dB ⊆ d ( A ∪ B ). Conversely suppose p / ∈ dA ∪ dB . Then there areopen neighborhoods U, V of P so that U ∩ A is meager and V ∩ B is meager. But then U ∩ V isan open neighborhood of p whose intersection with A ∪ B is meager, because ( U ∩ V ) ∩ ( A ∪ B ) ⊆ ( U ∩ A ) ∪ ( V ∩ B ), where the latter is a union of two meager sets and hence meager. KURATOWSKI CLOSURE-COMPLEMENT VARIANT WHOSE SOLUTION IS INDEPENDENT OF ZF 3 (e)
Since X is Polish, we may find a countable family of open sets { B i : i ∈ N } which comprisea basis for the topology of X . Form the subcollection C = { B i : B i ∩ A is meager } . For each x ∈ A − dA , since x is not a point of second category for A , we may find a neighborhood B i forwhich x ∈ B i and B i ∩ A is meager, so B i ∈ C . This shows A − dA ⊆ S B i ∈C B i ∩ A , so A − dA is asubset of a countable union of meager sets and hence meager. (f ) If A is meager, then dA = ∅ immediately from the definition of d . Conversely, if dA = ∅ , then A = A − dA and A is meager by (e). (g) By (b), ddA ⊆ kdA = dA . By (a), (d), (e), and (f), dA ⊆ d (( A − dA ) ∪ dA ) = d ( A − dA ) ∪ ddA = ddA . (h) Note that kA − ikA is a closed nowhere dense set and hence meager. So by (c), (d) and (f),we have dkA = d (( kA − ikA ) ∪ ikA ) = d ( kA − ikA ) ∪ dikA = ∅ ∪ kikA = kikA. (i) By (b), (g), and (h), kidA = kikdA = dkdA = ddA = dA . (cid:3) Lemma 4 (ZF+DC) . Let X be a Polish space, and A ⊆ X . Then the following are equivalent.(a) A has the Baire property.(b) dA − A is meager.(c) idcA = cdA .(d) idA = cdcA .(e) dA = cidcA .(f ) dcA = kcdA .Proof. (a ⇒ b) Assume A has the Baire property, and find U ⊆ X open so that M = A ∆ U is ameager set. We have A = M ∆ U , and therefore dA = d (( M − U ) ∪ ( U − M )) ∪ ∅ = d ( M − U ) ∪ d ( U − M ) ∪ d ( M ∩ U ) = ∅ ∪ d (( U − M ) ∪ ( M ∩ U )) = dU by Lemma 3 (d) and (f). So dA = dU ,and by applying the same argument, we obtain dcA = dcU , because M = ( cA )∆( cU ).Now we have dA − A = ( dA − A − dcA ) ∪ (( dA − A ) ∩ dcA ) ⊆ ( cA − dcA ) ∪ ( dA ∩ dcA )= ( cA − dcA ) ∪ ( dU ∩ dcU ) . The set cA − dcA is meager by Lemma 3 (e), and the set dU ∩ dcU ⊆ kU ∩ kcU = kU − U isnowhere dense. So dA − A is a subset of the union of two meager sets, hence meager. (b ⇒ a) Assume dA − A is meager. Set U = idA , so U is an open subset of X . We have U − A meager because U − A ⊆ dA − A . Also, A − U = ( A − dA ) ∪ ( dA − U ), where A − dA is meagerby Lemma 3 (e), and dA − U = dA − idA is closed nowhere dense, so A − U is meager. Therefore A ∆ U is meager, and we conclude A has the Baire property. (b ⇒ c) Assume dA − A is meager. Applying Lemma 3 (b), (c), (d), (e), (f), and (i), as well asthe DeMorgan’s law for interior/closure, we have M. P. COHEN, T. JOHNSON, A. KRAL, A. LI, AND J. SOLL idcA = id (( cA − dA ) ∪ ( cA ∩ dA ))= i [ d ( cA − dA ) ∪ d ( cA ∩ dA )]= i [ d ( cdA − A ) ∪ ∅ ∪ d ( dA − A )]= i [ d ( cdA − A ) ∪ d ( A − dA ) ∪ ∅ ]= id (( cdA − A ) ∪ ( cdA ∩ A ))= idcdA = ikcdA = ckidA = cdA. (c ⇒ d) Assume idcA = cdA . Note that idcA ∪ idA = idX = X by Lemma 3 (d), so idA ⊇ cidcA = kcdcA ⊇ cdcA . Conversely, cdA ∪ cdcA = idcA ∪ cdcA = i ( dcA ∪ cdcA ) = iX = X , so cdcA ⊇ ccdA = dA ⊇ idA . So idA = cdcA . (d ⇒ e) Assume idA = cdcA . Taking the closure of both sides, we have dA = kidA = kcdcA = cidcA . (e ⇒ f ) Assume dA = cidcA . Since dA ∪ dcA = d ( A ∪ cA ) = X by Lemma 3 (d), we have dcA ⊇ cdA , and since dcA is closed (Lemma 3 (b)) we also have dcA ⊇ kcdA . Conversely, we have cidcA ∪ kcdA = dA ∪ kcdA ⊇ dA ∪ cdA = X , so kcdA ⊇ ccidcA = idcA . Since kcdA is closed, wealso have kcdA ⊇ kidcA = dcA by Lemma 3 (i). So dcA = kcdA . (f ⇒ b) Assume dcA = kcdA . To show dA − A is meager, by Lemma 3 (f) it suffices to show that d ( dA − A ) = ∅ . We have d ( dA − A ) = d ( dA ∩ cA ) ⊆ ( ddA ∩ dcA ) by Lemma 3 (a). Therefore byassumption, d ( dA − A ) ⊆ dA ∩ kcdA , so id ( dA − A ) ⊆ idA ∩ ikcdA = idA ∩ ckidA = idA ∩ cdA ⊆ dA ∩ cdA = ∅ . Therefore d ( dA − A ) = kid ( dA − A ) = k ∅ = ∅ . (cid:3) Proof of Theorem 2.
We work in ZF+DC+PB. We denote by K the monoid of set operators gen-erated by k and i , and by KD the monoid of set operators generated by k , i , and d . Since i = ckc , K and KD are submonoids of KD . The proof of Kuratowski’s closure-complement theorem relieson observing that kiki = ki and ikik = ik , and therefore K = { e, i, k, ki, ik, iki, kik } ,where e denotes the identity operator. K is often called the monoid of even operators in the closure-complement problem. Using the tools of Lemma 3, we compute the (no more than seven) membersof K d : d , id , kd = d , kid = d , ikd = id , ikid = id , and kikd = d . So KD ⊇ K d = { d, id } . In fact,we have the following set equality: KD = { e, i, k, ki, ik, iki, kik, d, id } which is easily verified by using Lemma 3 to check that i KD ⊆ KD , k KD ⊆ KD , and d KD ⊆KD . So there are nine even operators in KD . Applying c to the left (or right) of KD yields ninemore operators, the odd operators, as depicted in Figure 1.The equalities in the last two entries in the table of Figure 1 hold because every set in X hasthe Baire property, allowing us to apply Lemma 4 universally. Noting that c KD = KD c consistsof the nine odd operators in the right column, we claim that KD = KD ∪ c KD , which proves KURATOWSKI CLOSURE-COMPLEMENT VARIANT WHOSE SOLUTION IS INDEPENDENT OF ZF 5
Even Operators Odd Operators e ci ci = kck ck = kiki cki = ikcik cik = kiciki ciki = kikckik ckik = kikcd cd = idcid cid = kcd = dc Figure 1.
Operators in KD = KD ∪ c KD .that KD ≤
18, as claimed. To see this, simply check the following set equalities which show that KD ∪ c KD is closed under multiplication from the left by the generating operators k , c , and d : k ( KD ∪ c KD ) = k KD ∪ kc KD = KD ∪ ci KD = KD ∪ c KD ; c ( KD ∪ c KD ) = c KD ∪ cc KD = KD ∪ c KD ; d ( KD ∪ c KD ) = d KD ∪ dc KD = KD ∪ cid KD = KD ∪ c KD .To prove the second statement of the theorem, we can take for example A = (1 , ∪ (2 , ∪ { } ∪ [(5 , ∩ Q ] ∪ [(6 , ∩ ( R − Q )]and see that application of the 9 even operators in KD yields 9 distinct sets as depicted in Figure2. Taking complements, we get 18 distinct sets from the 18 distinct operators in KD . eA (1 , ∪ (2 , ∪ { } ∪ [(5 , ∩ Q ] ∪ [(6 , ∩ ( R − Q )] iA (1 , ∪ (2 , kA [1 , ∪ { } ∪ [5 , kiA [1 , ikA (1 , ∪ (5 , ikiA (1 , kikA [1 , ∪ [5 , dA [1 , ∪ [6 , idA (1 , ∪ (6 , Figure 2.
Even operators applied to A . (cid:3) Vitali Sets and Distinguishing Words Under AC
Recall that a
Vitali set is a subset V ⊆ R consisting of exactly one representative from each cosetin the quotient group R / Q . Vitali sets can be constructed by invoking the Axiom of Choice, and donot have the Baire property. Proposition 5 (ZFC) . Let W ⊆ R be open. Then there exists a Vitali set V such that dV = kW . M. P. COHEN, T. JOHNSON, A. KRAL, A. LI, AND J. SOLL
Proof.
Let α be an arbitrary irrational real number, and let H = h Q , α i be the additive subgroupof R generated by Q and α , so H = { q + nα : q ∈ Q , n ∈ Z } . Let V be a set consisting of exactlyone representative from each coset of R /H . For each n ∈ Z , we define the sets P n = { v + nα + Q : v ∈ V } ⊆ R / Q and R n = S P n ⊆ R .We first claim that S n ∈ Z P n = R / Q . The left-to-right inclusion is by definition. For the right-to-left inclusion, consider an arbitrary coset x + Q in R / Q . There exists a unique element v ∈ V forwhich x + H = v + H , i.e. x − v ∈ H . Therefore we may write x − v = q + nα for some q ∈ Q andsome n ∈ Z . But then x − q = v + nα , whence x + Q = v + nα + Q ∈ P n . So R / Q ⊆ S n ∈ Z P n asclaimed.Moreover, the sets P n are pairwise disjoint. For if P n ∩ P m = ∅ , it means that there are v, w ∈ V for which v + nα + Q = w + mα + Q , i.e. v + nα = w + mα + q for some q ∈ Q . But then v − w = ( m − n ) α + q ∈ H , so v = w by construction of V . In turn, this implies ( m − n ) α = − q ∈ Q ,and hence n = m since α is irrational.The preceding two paragraphs imply that the family { P n : n ∈ Z } forms a partition of R / Q .Consequently, the sets R n = nα + R comprise a partition of R , and we conclude that each set R n is nonmeager in R .Next, let { B n : n ∈ Z } be a countable basis of open sets for the topology on W . For each n ∈ Z ,each coset v + nα + Q in P n is dense in R , and hence meets B n . So let V n be a set consisting ofexactly one element chosen from each intersection ( v + nα + Q ) ∩ B n ( v ∈ V ). Then V = S n ∈ Z V n consists of exactly one representative from each distinct coset in S n ∈ Z P n = R / Q , so V is a Vitaliset.If x ∈ kW , and U ⊆ R is an arbitrary open neighborhood of x , then there exists n ∈ Z with B n ⊆ U . But then V n ⊆ V ∩ B n ⊆ V ∩ U , and R n = S P n = S z ∈ V n S q ∈ Q ( z + q ) = S q ∈ Q ( V n + q ).Since R n is nonmeager, it follows that V n is nonmeager and hence V ∩ U is nonmeager. So x ∈ dV and we have shown kW ⊆ dV . Conversely, V ⊆ W so dV ⊆ kV ⊆ kW , and the proposition isproven. (cid:3) Proposition 6 (ZFC) . Let W ⊆ W ⊆ R such that W and W are both open. Then there existsa Vitali set V such that dV = kW and kV = kW .Proof. Using Proposition 5, start with a Vitali set V satisfying dV = kW . Let { v n : n ∈ N } be anarbitrary sequence of distinct elements in V , and let { C n : n ∈ N } be a countable basis of open setsfor the topology on W . For each n , let w n ∈ ( v n + Q ) ∩ C n . Define V = ( V − { v n : n ∈ N } ) ∪ { w n : n ∈ N } , so V is a Vitali set.Since V ∆ V is countable, hence meager, we have dV = dV = kW . We also have kV ⊆ kW since V ⊆ W , and kW ⊆ kV since V is dense in W by construction. (cid:3) Proof of Theorem 1.
Working in ZFC, Lemma 3 still holds, so KD consists of at least the 18 op-erators in KD ∪ c KD (see Figure 1). However, the identities in Lemma 4 do not apply to everysubset of X , and thus in general we do not have idc = cd , id = cdc , d = cidc , or dc = kcd . To seethat these equalities fail, we apply Proposition 6 and construct V a Vitali set satisfying dV = [8 , kV = [8 , cV has the following property: for every open set U in R , the intersection U ∩ cV contains a representative from each coset of Q (in fact infinitely many representatives). Thus KURATOWSKI CLOSURE-COMPLEMENT VARIANT WHOSE SOLUTION IS INDEPENDENT OF ZF 7 R ⊆ S q ∈ Q q + ( U ∩ cV ), so R is covered by countably many translates of U ∩ cV . This implies U ∩ cV is nonmeager.The preceding paragraph implies dcV = kcV = R . So V distinguishes additional operators in themonoid KD , as depicted in the table below: idcV = R cdV = R − [8 , idV = (8 , cdcV = ∅ dV = [8 , cidcV = ∅ dcV = R kcdV = R − (8 , KD = { e, i, k, ki, ik, iki, kik, d, id, c, ci, ck, cki, cik, ciki, ckik, cd, cid, dc, idc, cdc, cidc } .To verify the claim, one must check that KD is invariant under both left and right multiplicationby k , c , and d , and we leave the task to the reader using Lemma 3. So KD ≤ R which distinguishes all 22 operators, we give A = (1 , ∪ (2 , ∪ { } ∪ [(5 , ∩ Q ] ∪ [(6 , ∩ ( R − Q )] ∪ V ,where V is as in the first paragraph of the proof. (cid:3) Remark.
Examining the 22 operators in KD in Theorem 1, we may regard them as either even or odd depending on the number of instances of the c operator in the reduced word. So we find 11 evenoperators and 11 odd operators. However, in this case the monoid KD generated by k , i , and d doesnot yield all even operators, nor does either of the sets c KD or KD consist of all odd operators.(Contrast with the situation in the original closure-complement problem, and in Theorem 2.) Partial Orderings and Other Addenda
The monoid KD admits a natural partial ordering defined by the rule o ≤ o if and only if o A ⊆ o A for every set A ⊆ X . The partial ordering on K (the monoid generated by k and i )has been diagrammed by various authors; see for instance [5]. In general if o ≤ o then io ≤ io , ko ≤ ko , and co ≥ co . We observe also the following proposition. Proposition 7 (ZF+DC) . The following relations hold among even operators in KD .(a) d ≤ kik ;(b) iki ≤ cdc ;(c) cdc ≤ id ;(d) cdc ≤ cidc ;(e) ki ≤ cidc ; and(f ) cidc ≤ d .Proof. (a) Since d ≤ k , we have d = kid ≤ kik . (b) By (a) we have dc ≤ kikc , and hence cdc ≥ ckikc = iki . (c) Let A ⊆ X be arbitrary. If p ∈ cdcA , then p has an open neighborhood U for which U ∩ cA is meager. Given arbitrary x ∈ U and an arbitrary open neighborhood V of x , we can observe that( U ∩ V ) ∩ cA ⊆ U ∩ cA is meager, and hence ( U ∩ V ) ∩ A is nonmeager, because U ∩ V is nonmeager(being an open set). So V ∩ A is nonmeager, which implies x ∈ dA . Therefore U ⊆ dA which implies p ∈ idA and cdcA ⊆ idA . M. P. COHEN, T. JOHNSON, A. KRAL, A. LI, AND J. SOLL (d)
Since id ≤ d we have idc ≤ dc and therefore cdc ≤ cidc . (e) Apply k to the left side of the inequality in (b). (f ) Apply k to the left side of the inequality in (c). (cid:3) Combining the preceding inequalities with the known ordering on K , we obtain the partialordering on the even operators of KD presented in Figure 3. For each pair of even operators o , o ∈ KD not connected by an arrow in the diagram, the reader may verify that o A o A where A is one of the sets given in the proofs of Theorems 1 and 2. Thus the diagram is complete. i iki cdc id ikki dcidc kik ke i iki id ikki d kik ke Figure 3.
Left: the partial ordering on the 11 even operators of KD in ZFC. Right: the partial ordering on the 9 even operators of KD in ZF+DC+PB. Example 8 (Another ZF-Independent Problem) . We also consider the problem of the cardinality ofthe monoid
KFD = h k, c, f, d i generated by k , c , d , and the topological frontier operator f definedby f A = kA ∩ kcA for all sets A . As one would expect, the cardinality of this monoid also dependson axiomatic assumptions.The submonoid generated by k , c , and f has size ≤
34, as shown by Gaida and Eremenko in [4].This can be computed using the following identities: f f f = f f , f c = kf = f , f f k = f k , if k = 0, f kik = f ki , and f iki = f ik , where 0 denotes the “empty set operator” defined by 0 A = ∅ for every A . To this list we add the following three identities whose proofs we leave to the reader: df = kif , f id = f d , df k = 0. We are also interested in cardinality of the submonoid generated by k, i, f, d .We compute the following presentations and cardinalities.Axiom System Generators Cardinality List of ElementsZF+DC h k, i, f, d i { e, k, i, d, f, ik, f k, ki, f i, f d, id, if, f f,kif, kik, f ik, if k, iki, f ki, f if } ZF+DC+PB h k, c, f, d i { above } ∪ { c, ck, ci, cd, cf, cik, cf k, cki,cf i, cf d, cid, cif, cf f, ckif, ckik, cf ik, c , ciki, cf ki, cf if } ZFC h k, c, f, d i { above } ∪ { dc, idc, cdc, cidc, f dc, cf dc } The initial set A given in the proof of Theorem 1 is sufficient to distinguish the 46 operators in KFD . Remark (Suggestions for Further Projects) . Several variant problems remain open to be solved byan interested party. For example, how many distinct sets are obtainable using k , i , d , together withone or both of ∩ and ∪ ? (See [5] Section 4 for more information.)Also, it was shown by Kuratowski that it is possible to obtain infinitely many sets using k , c , andeither ∩ or ∪ . We believe replacing k with d should yield a finite answer and it may be interestingto compute.More broadly, the operator d is an example of a local function associated to a σ -ideal I on atopological space X . A general local function ℓ associated to I assigns to a set A ⊆ X the set ℓA consisting of all points p ∈ X for which every open neighborhood U of p satisfies U ∩ A / ∈ I . For d , KURATOWSKI CLOSURE-COMPLEMENT VARIANT WHOSE SOLUTION IS INDEPENDENT OF ZF 9 the σ -ideal in question is the family of meager subsets of X . It may be interesting to study variantsof the Kuratowski problem using local functions associated to other σ -ideals.Moreover, given a local function ℓ , the operator k ℓ defined by k ℓ A = A ∪ ℓA is an exampleof a Kuratowski closure operator, which generates a topology finer than the original. In fact thenew topology and the old are saturated in the sense that every open set in either has nonemptyinterior in the other. There exists some literature on variants of the Kuratowski problem in spacesequipped with multiple topologies (i.e. polytopological spaces ), including the special case of saturatedpolytopological spaces—see especially [1] and [3]. The creative reader may be able to craft interestingproblems by combining the machinery of local functions and polytopological spaces. References [1] T. Banakh, O. Chervak, T. Martynyuk, M. Pylypovych, A. Ravsky, and M. Simkiv,
Kuratowski monoids of n -topological spaces, Topological Algebra and its Applications 6, no. 1 (2018), 1–25.[2] M. Bowron,
Kuratowski’s Closure-Complement Cornucopia (2012). https://mathtransit.com/cornucopia.php [3] S. Canilang, M. P. Cohen, N. Graese, and I. Seong,
The closure-complement-frontier problem in saturated poly-topological spaces , preprint.[4] Yu. R. Gaida and A. ´E. Eremenko,
On the frontier operator in Boolean algebras with a closure , Ukr. Math. J.26.6 (1974), 806–809.[5] B. J. Gardner and M. Jackson,
The Kuratowski closure-complement theorem,
New Zealand J. Math. 38 (2008),9–44.[6] K. Kuratowski,
Sur l’operation A de l’Analysis Situs , Fundamenta Mathematicae 3 (1922), 182-199.[7] K. Kuratowski, Topology: Vol. 1, trans. J. Jaworowski, New York, Academic Press, 1966.[8] B. J. Pettis,
On continuity and openness of homomorphisms in topological groups , Ann. of Math. (2) 52, 1950,293–308.[9] C. Rosendal,
Automatic continuity of group homomorphisms , Bull. Symbolic Logic
15 (2) (2009), 184–214.[10] Shelah, S.,
Can you take Solovay’s inaccessible away? , Israel J. Math. 48(1) (1984), 1–47.[11] D. Sherman,
Variations on Kuratowski’s 14-set theorem , Amer. Math. Monthly, 117:2 (2010), 113–123.[12] Solovay, R.M.,
A model of set-theory in which every set of reals is Lebesgue measurable , Ann. Math. 92 (1970),1–56.
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