Prime Density and Classification of Macías Spaces over Principal Ideal Domains

Prime Densit y and Classification of Mac ´ ıas Spaces o v er Principal Ideal Domains Souvik Mandal a, ∗ and Ankur Sark ar a a Dep artment of Mathematics, Indian Institute of T e chnolo gy Madr as, Chennai 600036, T amil Nadu, India. Abstract Recen tly , the Mac ´ ıas top ology has been generalized ov er integral domains that are not fields, to furnish a top ological pro of of the infinitude of prime elemen ts under the assumption that the set of units is finite or not open. In this article, w e remov e this cardinalit y assumption completely by using the Jacobson radical. W e prov e that in any semiprimitive integral domain, the group of units is not op en in the Mac ´ ıas top ology . Consequen tly , for a principal ideal domain, this gives an equiv alence b et ween the triviality of the Jacobson radical, the density of the set of prime elements, and the group of units not being op en in the Mac ´ ıas top ology . F urthermore, we completely c haracterize when Mac ´ ıas spaces o ver different infinite principal ideal domains are homeomorphic in terms of cardinalities of certain subsets of the domains. As an application w e resolve an op en problem concerning homeomorphism of Mac ´ ıas spaces ov er countably infinite semiprimitive principal ideal domains. Keywor ds: Mac ´ ıas top ology , Principal ideal domain, Jacobson radical, Semiprimitiv e ring, Prime densit y , Infinitude of primes, Homeomorphism classification. 2020 Mathematics Subje ct Classific ation: Primary 54A05, 13F10, 16N20; Secondary 13G05. 1 In tro duction The in tersection of top ology and n umber theory has a rich history , most famously inaugurated b y F ursten- b erg’s 1955 top ological pro of of the infinitude of prime num b ers [ 1 ]. F urstenberg’s top ology on Z , gen- erated by arithmetic progressions, is Hausdorff and metrizable. In con trast, Golomb [ 2 ] and Kirch [ 3 ] explored top ologies on N that are connected but not Hausdorff, emphasising the arithmetic nature of op en sets. In 2024, Mac ´ ıas introduced a coarser top ology on N generated by sets of integers coprime to a fixed in teger [ 4 ]. This notion was subsequently extended to arbitrary in tegral domains R , resulting in the top ology now kno wn as the Mac ´ ıas top ology [ 5 ]. A primary application of this top ology in [ 5 ] was to provide a top ological pro of of the infinitude of prime elements in principal ideal domains under the assumption that the group of units is finite. In this article, w e demonstrate that the aforementioned finiteness hypothesis can be completely remo ved by using the Jacobson radical. In fact, our contributions are as follows: (i) W e dev elop a prime supp ort framework for the basic op en sets of the Mac ´ ıas top ology ov er unique factorization domains, bridging the arithmetic and top ological structures (see Section 2 ). This framew ork serves as a key to ol in the pro ofs of b oth Theorem 3.1 and Theorem 4.1 . (ii) W e pro ve that an in tegral domain that is not a field is semiprimitive (that is, has a trivial Jacobson radical) if and only if its group of units is not op en in the Mac ´ ıas top ology . F or principal ideal domains, w e further establish that this condition is equiv alent to tw o additional prop erties: the top ological densit y of the set of asso ciate classes of primes, and the infinitude of prime elements (see Section 3 ). (iii) W e provide a complete classification of Mac ´ ıas spaces ov er infinite principal ideal domains, proving that tw o suc h spaces are homeomorphic if and only if their groups of units hav e the same cardinality and their sets of asso ciate classes of primes hav e the same cardinality , thereby resolving an op en Problem 4.1 p osed in [ 5 ] (see Section 4 ). ∗ Corr esp onding author. E-mail addresses: ma22d014@smail.iitm.ac.in , ssouvik.xyz@gmail.com (S. Mandal); ankurimsc@gmail.com (A. Sark ar). 1 S. MAND AL AND A. SARKAR 2 Preliminaries Throughout this pap er, R denotes a comm utative in tegral domain with identit y 1  = 0 that is not a field, unless otherwise stated; in particular, R is necessarily infinite. F or any subset X ⊆ R , we write X 0 = X \ { 0 } for the set of its non-zero elements. In particular, R 0 denotes the punctured ring R \ { 0 } , and U ( R ) denotes its group of units. Tw o elements a, b ∈ R are called asso ciates , written a ∼ b , if a = ub for some u ∈ U ( R ). W e write P R for the set of asso ciate classes of prime elements of R ; for a prime p , its class is denoted by [ p ] ∈ P R . W e denote by Fin( P R ) the collection of all finite subsets of P R . Definition 2.1 ([ 5 ]) . F or each k ∈ R 0 , define σ 0 k := { s ∈ R 0 : ⟨ k ⟩ + ⟨ s ⟩ = R } , where ⟨ k ⟩ denotes the principal ideal generated b y k . Let, B = { σ 0 k : k ∈ R 0 } . F ollo wing [ 5 , Theorem 2.1] the set B forms a basis for some top ology on R . The Mac ´ ıas sp ac e over R , denoted by M ( R ), is the top ological space ( R 0 , τ R 0 ), where τ R 0 is the top ology generated b y the basis B . V arious top ological prop erties of the Maci´ as space M ( R ) hav e b een studied in [ 5 ]. In particular, it is not a Hausdorff space, and hence is not metrizable. Note that R 0 is itself a basic op en set since, for an y unit u of R , we hav e σ 0 u = R 0 . T o connect the algebraic structure with the topological constructions, we introduce the notion of prime supp ort, applicable whenev er R is a unique factorization domain. Definition 2.2. Let R b e a unique factorization domain. F or any α ∈ R 0 , the prime supp ort of α , denoted supp( α ), is the set of asso ciate classes of prime elemen ts that divide α . In particular, supp( u ) = ∅ for every unit u ∈ U ( R ). R emark 2.3 . When R is a unique factorization domain, ev ery non-zero non-unit admits a unique factor- ization into finitely man y prime eleme n ts, ensuring that supp( α ) ∈ Fin( P R ) for all α ∈ R 0 . W e note that in an in tegral domain where every non-unit factors into finitely many irreducibles but these irreducibles are not necessarily prime, the set supp( α ) may b e empty even for non-units. The following pair of lemmas establishes the relationship b et ween prime supp ort and the basic op en sets of the Mac ´ ıas top ology in a unique factorization domain. Lemma 2.4. L et R b e a unique factorization domain. F or any s, k ∈ R 0 , if s ∈ σ 0 k , then supp( s ) ∩ supp( k ) = ∅ . Pr o of. Since s ∈ σ 0 k , there exist x, y ∈ R with xs + y k = 1. If a prime p divides b oth s and k , then p divides xs + y k = 1, which is a contradiction. Hence supp( s ) ∩ supp( k ) = ∅ . R emark 2.5 . The conv erse of Lemma 2.4 fails in general. In Z [ x ], the elements 2 and x are prime with supp(2) ∩ supp( x ) = ∅ , yet ⟨ 2 ⟩ + ⟨ x ⟩ = ⟨ 2 , x ⟩ ⊊ Z [ x ], so 2 / ∈ σ 0 x . Establishing the con verse requires a stronger algebraic structure, such as a principal ideal domain. Lemma 2.6. L et R b e a princip al ide al domain. F or any s, k ∈ R 0 , if supp( s ) ∩ supp( k ) = ∅ , then s ∈ σ 0 k . Pr o of. Let d = gcd( s, k ), which exists since R is a principal ideal domain. If d is not a unit, then there exists a prime p dividing d . Consequently , p divides b oth s and k , so that [ p ] ∈ supp( s ) ∩ supp( k ), con tradicting the hypothesis. Hence d is a unit. Therefore ⟨ s ⟩ + ⟨ k ⟩ = ⟨ d ⟩ = R , and thus s ∈ σ 0 k . Com bining Lemmas 2.4 and 2.6 , we obtain the follo wing equiv alence for principal ideal domains. Corollary 2.7. L et R b e a princip al ide al domain. F or any r , s ∈ R 0 , one has r ∈ σ 0 s if and only if supp( r ) ∩ supp( s ) = ∅ . In p articular, the b asic op en sets of M ( R ) dep end exclusively on prime supp ort. W e conclude this section with a characterization of units that is central to the classification theorem in Section 4 . Lemma 2.8. In the top olo gic al sp ac e M ( R ) , an element x ∈ R 0 is a unit if and only if the closur e of the singleton { x } is the entir e sp ac e, that is, { x } = R 0 . 2 Prime Densit y and Classification of Mac ´ ıas Spaces ov er Principal Ideal Domains Pr o of. If x is a unit, then ⟨ x ⟩ + ⟨ k ⟩ = R for every k ∈ R 0 , so x ∈ σ 0 k for all k ∈ R 0 . It follows that every basic op en set con taining any p oin t y ∈ R 0 also contains x . Therefore { x } = R 0 . Con versely , supp ose { x } = R 0 . Then x b elongs to every basic op en set. In particular, x ∈ σ 0 x . This implies that ⟨ x ⟩ + ⟨ x ⟩ = R . Hence ⟨ x ⟩ = R , forcing x ∈ U ( R ). This completes the pro of. R emark 2.9 . The forwar d implication of Lemma 2.8 w as obtained indep enden tly by Mac ´ ıas and Ortiz in [ 6 , Theorem 5.5] via the closure formula for the Jacobson radical of an ideal. 3 Semiprimitivit y and Prime Density In [ 5 , Subsection 3.16], the assumption that U ( R ) is finite w as used to ensure that it is not op en in M ( R ). In Theorem 3.3 w e characterize this top ological condition with an algebraic one with no cardinalit y condition. Recall that the Jac obson r adic al J ( R ) of a commutativ e ring R is the intersection of all its maximal ideals, and that R is called semiprimitive if J ( R ) = { 0 } . In this section we prov e the following Theorem. Theorem 3.1. L et R b e a princip al ide al domain that is not a field. The fol lowing ar e e quivalent: (a) U ( R ) is not op en in M ( R ) . (b) The set of prime elements P R is dense in M ( R ) . (c) The set of prime elements P R is infinite. (d) R is semiprimitive. R emark 3.2 . The class of semiprimitive principal ideal domains is extensive. Notable examples include the ring of integers Z , p olynomial rings F [ x ] ov er an arbitrary field F , and rings of integers O K of n umber fields K with class n umber one—suc h as the Gaussian integers Z [ i ], the Eisenstein integers Z [ ω ], and the real quadratic ring Z [ √ 2]. More generally , Theorem 3.1 implies that any principal ideal domain with infinitely many non-asso ciate prime elements is semiprimitive. In particular, the rings of S -integers O K,S with trivial S -class group and the lo calizations Z [ S − 1 ] for an y finite set of rational primes S furnish further examples. Before proving the Theorem 3.1 we hav e the following Theorem 3.3 and Lemma 3.4 which we use for Pro ving Theorem 3.1 . Theorem 3.3. L et R b e an inte gr al domain that is not a field. The ring R is semiprimitive if and only if the gr oup of units U ( R ) is not an op en set in M ( R ) . Pr o of. First, assume R is semiprimitive. Supp ose, for the sake of contradiction, that U ( R ) is op en in M ( R ). Since 1 ∈ U ( R ), there exists some k ∈ R 0 suc h that σ 0 k ⊆ U ( R ). Observ e that k m ust b e a non-unit: if k were a unit, then σ 0 k = R 0 b y [ 5 , Theorem 2.2], forcing U ( R ) = R 0 , whic h con tradicts the assumption that R is not a field. Thus, k is a non-unit. Let m b e an arbitrary maximal ideal of R . If k / ∈ m , the maximalit y of m yields ⟨ k ⟩ + m = R , implying there exist elements x ∈ R and s ∈ m suc h that xk + s = 1. Consequen tly , ⟨ k ⟩ + ⟨ s ⟩ = R . Since k is not a unit, s  = 0, meaning s ∈ σ 0 k . By our assumption, s ∈ U ( R ). Ho wev er, a prop er ideal cannot contain a unit, resulting in a contradiction. Therefore, k ∈ m for every maximal ideal m , yielding k ∈ J ( R ). Since R is semiprimitive, J ( R ) = { 0 } , whic h forces k = 0, contradicting k ∈ R 0 . Thus, U ( R ) is not op en. Con versely , assume R is not semiprimitiv e. W e will show that U ( R ) is op en in M ( R ). Since J ( R )  = { 0 } , there exists a non-zero element k ∈ J ( R ). Consider the basic open set σ 0 k . Let s b e an arbitrary elemen t in σ 0 k , which by definition means ⟨ k ⟩ + ⟨ s ⟩ = R . If s were a non-unit, then by Krull’s Theorem s is contained in some maximal ideal m of R . F urthermore, as k ∈ J ( R ), the element k is contained in all maximal ideals of R , including m . This implies that ⟨ k ⟩ + ⟨ s ⟩ ⊆ m ⊊ R , which directly contradicts ⟨ k ⟩ + ⟨ s ⟩ = R . Hence, the elemen t s must b e a unit. As this holds for every s ∈ σ 0 k , we deduce that σ 0 k ⊆ U ( R ). Thus we hav e u ∈ σ 0 k ⊆ U ( R ) for all u ∈ U ( R ). Thus U ( R ) is an op en set. This establishes the equiv alence. In [ 5 , Theorem 3.18, Corollary 3.21], it is sho wn that if a principal ideal domain R p ossesses a finite (and hence non-op en) unit group U ( R ), then the set P R m ust b e infinite. How ever, Lemma 3.4 demonstrates that this implication holds more generally for any unique factorization domain. 3 S. MAND AL AND A. SARKAR Lemma 3.4. L et R b e a unique factorization domain that is not a field. If P R is finite, then U ( R ) is op en in M ( R ) and P R is not dense in M ( R ) . Pr o of. Let { p 1 , . . . , p n } b e a complete set of representativ es for P R and set α = p 1 · · · p n . If x ∈ σ 0 α , then b y Lemma 2.4 , supp( x ) ∩ supp( α ) = ∅ . Since supp( α ) = P R , the element x admits no prime divisors and therefore is a unit. Hence σ 0 α ⊆ U ( R ), and since 1 ∈ σ 0 α , the set U ( R ) is op en. Moreo ver, every prime is a non-unit, so σ 0 α ∩ P R = ∅ . Hence P R is not dense. R emark 3.5 . The conv erse of Lemma 3.4 do es not hold for unique factorization domains in general as explained in Remark 3.6 . Pr o of of The or em 3.1 . (b) ⇒ (a) . If P R is dense, then every basic open set σ 0 k con tains a prime element. Since primes are non-units, no basic op en set is contained in U ( R ), so U ( R ) is not op en. (a) ⇒ (c) . This is the contrapositive of Lemma 3.4 : if P R w ere finite, then U ( R ) would b e op en. (c) ⇒ (b) . Let σ 0 k b e an arbitrary non-empty basic op en set. Since R is a principal ideal domain, k has only finitely many prime divisors. Because P R is infinite, there exists a prime p with [ p ] / ∈ supp( k ). By Corollary 2.7 , p ∈ σ 0 k , and hence σ 0 k ∩ P R  = ∅ . (d) ⇔ (a) F ollo ws from Theorem 3.3 . R emark 3.6 . A natural question arises as to whic h implications of Theorem 3.1 extend b eyond principal ideal domains. W e in vestigate this for unique factorization domains. (i) The equiv alence (d) ⇔ (a) holds for any integral domain. This follows from Theorem 3.3 . (ii) The implication (a) ⇒ (c) holds for any unique factorization domain. This follo ws from Lemma 3.4 . (iii) The implication (b) ⇒ (a) holds for any integral domain. The same argument as in the proof of (b) ⇒ (a) in Theorem 3.1 applies. (iv) F or unique factorization domains, the implications (c) ⇒ (b) , (c) ⇒ (a) , and (c) ⇒ (d) fail in general. F or this consider the formal p o wer series ring R = C [[ x, y ]]. This ring is an unique factorization domain (b eing a regular lo cal ring), and it p ossesses infinitely many pairwise non-asso ciate primes, including x , y , and x − cy n for each c ∈ C × and n ≥ 1. Thus (c) is satisfied. Ho wev er, R is a lo cal ring with the unique maximal ideal m = ⟨ x, y ⟩ , so that J ( R ) = m  = { 0 } . Therefore the statement (d) fails. T o verify the failure of (a) and (b) , observ e that R / ⟨ x ⟩ ∼ = C [[ y ]], which is a lo cal domain whose units are precisely the p o wer series with non-zero constant term. Consequently , s ∈ σ 0 x if and only if the image of s in C [[ y ]] is a unit, which o ccurs precisely when s has a non-zero constant term i.e., when s ∈ U ( R ). Hence σ 0 x = U ( R ), and U ( R ) is op en, so (a) fails. Since every prime elemen t of R lies in m and therefore has zero constant term, we obtain σ 0 x ∩ P R = ∅ , so (b) fails as well. 4 Classification of the Mac ´ ıas Space ov er Principal Ideal Do- mains In this section, we establish necessary and sufficient algebraic conditions for the Mac ´ ıas spaces of tw o principal ideal domains to be homeomorphic. In [ 5 , Problem 4.1], Mac ´ ıas p oses the follo wing open problem. Problem ([ 5 ]) . L et R and S b e c ountably infinite semiprimitive inte gr al domains. De cide whether M ( R ) and M ( S ) ar e home omorphic. W e resolve this problem by providing a complete classification of the Mac ´ ıas space of any infinite principal ideal domain, up to homeomorphism in Corollary 4.2 . Corollary 4.3 answers the Problem 4.1 raised by Mac ´ ıas in [ 5 ]. In fact, More generally , we prov e the following Classification Theorem. Theorem 4.1. L et R and S b e infinite princip al ide al domains that ar e not fields. Then the top olo gic al sp ac es M ( R ) and M ( S ) ar e home omorphic if and only if | U ( R ) | = | U ( S ) | and |P R | = |P S | . Corollary 4.2. L et R and S b e c ountably infinite semiprimitive princip al ide al domains that ar e not fields. Then M ( R ) and M ( S ) ar e home omorphic if and only if | U ( R ) | = | U ( S ) | . 4 Prime Densit y and Classification of Mac ´ ıas Spaces ov er Principal Ideal Domains Pr o of. By Theorem 3.1 , b oth P R and P S are infinite. Since R and S are coun tably infinite, b oth prime sets are countably infinite, so |P R | = |P S | = ℵ 0 . The result now follows from Theorem 4.1 . Corollary 4.3. Ther e exist c ountably infinite semiprimitive inte gr al domains R and S such that M ( R ) and M ( S ) ar e not home omorphic. Pr o of. T ake R = Z and S = F 2 [ x ]. Both are countably infinite semiprimitive principal ideal domains, but | U ( Z ) | = 2 while | U ( F 2 [ x ]) | = 1. By Corollary 4.2 , M ( Z ) and M ( F 2 [ x ]) are not homeomorphic. In order to prov e Theorem 4.1 , we require the closure structure of singletons in a principal ideal domain. W e hav e the following lemma from [ 5 , Corollaries 3.2, 3.5] to serve this purp ose. Lemma 4.4. L et R b e a princip al ide al domain and let p ∈ R b e a prime element. Then { p } = ⟨ p ⟩ 0 . Mor e gener al ly, if x ∈ R 0 \ U ( R ) admits the prime factorization x = u Q k i =1 p a i i with u ∈ U ( R ) , a i ≥ 1 , and the p i p airwise non-asso ciate primes, then { x } = k \ i =1 ⟨ p i ⟩ 0 . R emark 4.5 . It is w orth noting that Lemma 4.4 does not, in general, hold for unique factorization domains. T o observe this, consider the formal p ow er series ring R = C [[ x, y ]], which is a lo cal unique factorization domain with the unique maximal ideal m = ⟨ x, y ⟩ . Note that x and y are distinct, non- asso ciate prime elements in R . If the form ula in Lemma 4.4 holds, then { x } = ⟨ x ⟩ 0 , and consequently y / ∈ { x } . How ever, supp ose σ 0 k is any basic op en set con taining y . Then ⟨ k ⟩ + ⟨ y ⟩ = R . Since y is a non-unit, y ∈ m . If k is also a non-unit, then k ∈ m , forcing ⟨ k ⟩ + ⟨ y ⟩ ⊆ m ⊊ R , whic h contradicts ⟨ k ⟩ + ⟨ y ⟩ = R . Thus, k must b e a unit, which implies that σ 0 k = R 0 . Consequently , the only basic op en set containing y is the entire space R 0 , which trivially contains x . This shows that y ∈ { x } , and hence { x }  = ⟨ x ⟩ 0 . No w we introduce the notion of maximal prop er single closure in Mac ´ ıas Space, which will b e used in the pro of of Theorem 4.1 . Definition 4.6. A subset C ⊊ R 0 is called a maximal pr op er singleton closur e in M ( R ) if there exists x ∈ R 0 with C = { x } and no singleton closure prop erly contains C in R 0 . W e denote by C R the collection of all such subsets. The remainder of this section is devoted to the pro of of Theorem 4.1 . Pr o of of The or em 4.1 . W e establish each direction separately . Let H : M ( R ) → M ( S ) b e a homeomorphism. W e show that | U ( R ) | = | U ( S ) | and |P R | = |P S | . Since H is a homeomorphism, we hav e { H ( x ) } = H  { x }  . Hence, by Lemma 2.8 , if x ∈ U ( R ), then { H ( x ) } = H ( R 0 ) = S 0 . It follows that H ( x ) ∈ U ( S ) b y the same lemma. Thus H ( U ( R )) ⊆ U ( S ). Applying the same argumen t to H − 1 sho ws that H − 1 ( U ( S )) ⊆ U ( R ). Therefore H ( U ( R )) = U ( S ), and since H is a bijection, | U ( R ) | = | U ( S ) | . W e note from Lemmas 2.8 and 4.4 , the p ossible singleton closures in M ( R ) are: • { x } = R 0 when x is a unit, and • { x } = T k i =1 ⟨ p i ⟩ 0 when x is a non-unit with supp( x ) = { [ p 1 ] , . . . , [ p k ] } . Among the prop er singleton closures (those strictly contained in R 0 ), the maximal ones under set in- clusion are those of the form ⟨ p ⟩ 0 for a single prime p . T o see this, note that if k > 1, then for each j w e hav e the strict inclusion T k i =1 ⟨ p i ⟩ 0 ⊊ ⟨ p j ⟩ 0 , since an element divisible only by p j (and not by p i for i  = j ) lies in ⟨ p j ⟩ 0 but not in T i ⟨ p i ⟩ 0 . The preceding analysis shows that the collection C R in tro duced in Definition 4.6 can b e written as C R =  ⟨ p ⟩ 0 : [ p ] ∈ P R  , establishing a canonical bijection b et ween C R and P R , since tw o prime elemen ts in a principal ideal domain generate the same ideal if and only if they are asso ciates. Since H preserves closures and set inclusion, it sends maximal prop er singleton closures in M ( R ) to maximal prop er singleton closures in M ( S ). That is, H induces a bijection C R → C S . Comp osing with the canonical bijections C R ↔ P R and C S ↔ P S yields |P R | = |P S | . 5 S. MAND AL AND A. SARKAR Con versely , we assume | U ( R ) | = | U ( S ) | and |P R | = |P S | . W e show that M ( R ) is homeomorphic to M ( S ). Fix a bijection φ : P R → P S . This naturally induces a bijection ˆ φ : Fin( P R ) → Fin( P S ). Since R is a principal ideal domain (hence a unique factorization domain), every element x ∈ R 0 has a well-defined prime supp ort supp( x ) ∈ Fin( P R ). This induces a partition F = { F A : A ∈ Fin( P R ) } of R 0 , where F A := { x ∈ R 0 : supp( x ) = A } . Similarly , with G B := { y ∈ S 0 : supp( y ) = B } , the collection G = { G B : B ∈ Fin( P S ) } partitions S 0 . F or any non-empty A = { [ p 1 ] , . . . , [ p k ] } ∈ Fin( P R ), Lemma 4.4 implies that each element of F A can b e written uniquely as x = u p a 1 1 · · · p a k k , with u ∈ U ( R ) and in tegers a i ≥ 1 for all i . The num b er of choices for the unit u is | U ( R ) | , and the n umber of choices for the exp onen t tuple ( a 1 , . . . , a k ) ∈ N k is | N k | = ℵ 0 . Since R is a principal ideal domain (and hence a unique factorization domain), distinct choices pro duce distinct elements, yielding | F A | = | U ( R ) | × ℵ 0 , for every non-empty A ∈ Fin( P R ) . Applying an analogous argument to the principal ideal domain S yields | G B | = | U ( S ) | × ℵ 0 for ev ery non-empt y B ∈ Fin( P S ). W e note that | F ∅ | = | U ( R ) | and | G ∅ | = | U ( S ) | . Since | U ( R ) | = | U ( S ) | , it follows that | F ∅ | = | G ∅ | . Moreo ver, for every non-empty A ∈ Fin( P R ), we hav e | F A | = | U ( R ) | × ℵ 0 = | U ( S ) | × ℵ 0 = | G ˆ φ ( A ) | . By the Axiom of Choice, we fix a bijection h A : F A → G ˆ φ ( A ) for each A ∈ Fin( P R ). As the collections F and G partition R 0 and S 0 in to pairwise disjoint subsets resp ectiv ely , we get a well-defined global bijection H : R 0 − → S 0 , H ( x ) := h supp( x ) ( x ) . By construction, supp( H ( x )) = ˆ φ (supp( x )) for every x ∈ R 0 . Now to show that H is a homeomorphism, it suffices to verify that H and H − 1 eac h map basic op en sets to basic op en sets. Let r ∈ R 0 . By Corollary 2.7 , the basic op en set σ 0 r is given by σ 0 r = { x ∈ R 0 : supp( x ) ∩ supp( r ) = ∅} . Since supp( H ( x )) = ˆ φ (supp( x )) and ˆ φ is a bijection, we hav e H ( σ 0 r ) =  H ( x ) ∈ S 0 : supp( x ) ∩ supp( r ) = ∅  =  y ∈ S 0 : ˆ φ − 1 (supp( y )) ∩ supp( r ) = ∅  =  y ∈ S 0 : supp( y ) ∩ ˆ φ (supp( r )) = ∅  = σ 0 s , where s ∈ S 0 is any element with supp( s ) = ˆ φ (supp( r )). Thus H sends basic op en sets in M ( R ) to basic op en sets in M ( S ). By symmetry (replacing φ with φ − 1 ), H − 1 also sends basic op en sets to basic op en sets. Therefore H is a homeomorphism. This completes the pro of. 5 Concluding Remarks It is worth emphasizing that the principal ideal domain hypothesis is crucial to the methods of this article, since it simultaneously guarantees tw o key algebraic prop erties: Krull dimension one and a trivial ideal class group. The Krull dimension one condition, under which every nonzero prime ideal is maximal, guarantees that the sum of the ideals generated by tw o elements with disjoint prime supp orts is the entire ring (Lemma 2.6 ). This prop ert y ensures that the closure of a singleton decomp oses as the intersection of the principal ideals generated b y its prime factors (Lemma 4.4 ); as noted in Remark 4.5 , this decomp osition fails in higher-dimensional unique factorization domains. On the other hand, the triviality of the ideal class group guarantees unique factorization into prime elemen ts rather than just into prime ideals. This prop ert y ensures that the prime supp ort framew ork 6 Prime Densit y and Classification of Mac ´ ıas Spaces ov er Principal Ideal Domains in tro duced in Section 2 is well-defined and underlies the cardinalit y computations in the pro of of Theo- rem 4.1 , which in turn form the basis for the homeomorphism classification. While Theorem 3.1 and Theorem 4.1 are fully resolved for principal ideal domains, extending these results to broader classes of one-dimensional domains where unique factorization of elemen ts ma y fail remains a substantial challenge. W e p ose the followin g op en problem. Problem. If R is a De dekind domain (which ne c essarily has Krul l dimension one) with a non-trivial ide al class gr oup, how do es the structur e of the class gr oup influenc e the home omorphism typ e of M ( R ) ? Ac knowledgemen t. The first author is grateful for financial supp ort in the form of Prime Minister’s Researc h F ellowship, Gov ernment of India (PMRF/2502403). The second author was supp orted by the Cen tre for Op erator Algebras, Geometry , Matter and Spacetime, Ministry of Education, Gov ernmen t of India through Indian Institute of T echnology Madras (Pro ject no. SB22231267MAETW O008573). References [1] H. F urstenberg. On the infinitude of primes. Amer. Math. Monthly , 62:353, 1955. https://doi. org/10.2307/2307043 . [2] S. W. Golomb. A connected top ology for the integers. Amer. Math. Monthly , 66:663–665, 1959. https://doi.org/10.2307/2309340 . [3] A. M. Kirc h. A countable, connected, lo cally connected Hausdorff space. A mer. Math. Monthly , 76:169–171, 1969. https://doi.org/10.2307/2317265 . [4] J. Mac ´ ıas. Another top ological pro of of the infinitude of prime num b ers. Inte gers , 24:Paper No. A47, 4, 2024. https://doi.org/10.5281/zenodo.11221653 . [5] J. Mac ´ ıas. 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