@ Appl. Gen. Topol. 22, no. 1 (2021), 149-167doi:10.4995/agt.2021.14422 © AGT, UPV, 2021 On sheaves of Abelian groups and universality S. D. Iliadis ∗ and Yu. V. Sadovnichy Moscow State University (M.V. Lomonosov), Moscow Center of Fundamental and Applied Math- ematics (s.d.iliadis@gmail.com,sadovnichiy.yu@gmail.com) Communicated by D. Georgiou Abstract Universal elements are one of the most essential parts in research fields, investigating if there exist (or not) universal elements in different classes of objects. For example, classes of spaces and frames have been studied under the prism of this universality property. In this paper, studying classes of sheaves of Abelian groups, we construct proper universal el- ements for these classes, giving a positive answer to the existence of such elements in these classes. 2010 MSC: 14F05; 18F20; 54B40. Keywords: sheaves; universal sheaves; universal spaces; containing spaces; saturated classes of spaces. 1. Introduction and Preliminaries The notion of “universal object” is considered in many branches of Mathe- matics. The problem of the existence of such objects is naturally arised when- ever a new category of objects is appeared. Especially for the branch of Topol- ogy, the problem of the existence of universal elements in different classes of topological spaces was considered at the first steps of its development. Now, in the bibliography there are lots of papers concerning universal objects. Many of them are indicated in the book [9]. In the paper [7] and in the above mentioned book, a method of construction of so-called Containing Spaces is developed. This method can be used for the ∗Corresponding author. Received 29 September 2020 – Accepted 20 December 2020 http://dx.doi.org/10.4995/agt.2021.14422 S .D. Iliadis and Yu. V. Sadovnichy construction of universal objects in different categories. Such categories are, for example, topological spaces (with different dimension invariants)(see [3], Chapter 3 of [9]), separable metric spaces (see Chapter 9 of [9], [10], [12], [13], [15]), mappings (see [8], Chapter 6 of [9], [10], [11]), topological groups ([11], [14], [17]), G-spaces (see Chapter 7 of [9], [10], [11], [17], [18]) and frames (see [2], [4], [5], [6], [16]). In the present paper we use this method for construction of universal objects in the category of sheaves of Abelian groups, which play an important role in the study of cohomology theories of general topological spaces. General notation and assumptions. An ordinal is considered as the set of all smaller ordinals. A cardinal is identified with the least ordinal of this cardinality. By τ we denote a fixed infinite cardinal. By F we denote the set of all non-empty finite subsets of τ. The symbol ≡ in a relation means that one or both sides of the relation are new notations. All spaces are assumed to be T0-spaces of weight ≤ τ. An equivalent relation on a set X is considered as a subset of X × X. 1.1 On the sheaves. We consider the notion of a sheaf according to [1]. A sheaf of Abelian groups is a triad (A, π, X) satisfying the following conditions: (i) A and X are topological spaces and π is a map of A onto X. (ii) π is a local homeomorphism, that is each point a ∈ A has an open neighbourhood V in A such that the restriction of π on V is a homeomorphism of V onto an open subset of X; (iii) for each point x ∈ X the set Ax ≡ π −1(x), which is called fiber of A in x, is an Abelian group; (iv) the group operations are continuous. (This condition means the follow- ing. Let A ⊠ A be the set of all pairs (a, b) ∈ A × A such that π(a) = π(b). Then, the mapping ̟A : A ⊠ A → A for which ̟A(a, b) = a + b is continuous. Similarly, the mapping iA : A → A for which iA(a) = −a is continuous.) Below we give some well-known notions of sheaves and introduce some no- tations, which will be used in the paper. Let p1 ≡ (A1, π1, X1) and p2 ≡ (A2, π2, X2) be two sheaves. A continuous mapping f of A1 into A2 is called homomorphism if the restriction of f onto each fiber of A1 is a homomorphism of this fiber into a fiber of A2. The unique mapping g of X1 into X2 satisfying the relation g ◦ π1 = π2 ◦ f is called induced by f. The homomorphism f is called isomorphism (or embedding) of p1 into p2 if f and the induced mapping g are embeddings. The isomorphism f of p1 into p2 is called proper if for each x ∈ X1 the restriction of f onto the fiber A1,x of A1 in x maps A1,x onto the fiber A2,g(x) of A2 in g(x). The sheaves p1 and p2 are called isomorphic if there exists an isomorphism of p1 onto p2. Let (A, π, X) be a sheaf and U a non-empty subset of X. A continuous mapping s : U → A, for which the maping π ◦ s is the identical mapping of U, is called © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 150 On sheaves of Abelian groups and universality section of A on U. We shall consider sections on the open subsets of X and the set of all such sections will be denoted by A(sec). The set of all sections on an open subset U of X will be denoted by A(sec)(U). The set A(sec)(U) is an Abelian group under the pointwise operations. For two subsets U and V such that U ⊂ V we denote by rAU,V the mapping of A(sec)(V ) into A(sec)(U) by setting rAU,V (s) = s|U for each s ∈ A(sec)(V ). The section s|U will be called restriction of the section s and the mapping rAU,V restriction mapping. Obviously, rAU,V is a homomorphism of the group A(sec)(V ) into the group A(sec)(U). Moreover, if U, V, W are subsets of X and U ⊂ V ⊂ W , then rAU,V ◦ r A V,W = r A U,W . For each section s : U → A the set dom(s) ≡ U is called the domain of s and the set ran(s) ≡ s(U) is called the range of s. For each subset B ⊂ A(sec) and x ∈ X we put dom(B) = {dom(s) : s ∈ B}, ran(B) = {ran(s) : s ∈ B} and dom(B)(x) = {dom(s) : s ∈ B, x ∈ dom(s)}. Also, for each non-empty subset U ⊂ X, we put B(U) = {s ∈ B : dom(s) = U}. We note that for each s ∈ A(sec) the set ran(s) is an open subset of A and that the set ran(A(sec)) is a base for the open subsets of A. The set dom(A(sec))(x) is directed by inclusion “ ⊂ ”. Thus, for each x ∈ X we have a direct spectrum of Abelian groups ΣAx ≡ {A(sec)(U), r A U,V , dom(A(sec))(x)}, where U, V ∈ dom(A(sec))(x) with U ⊂ V and rAU,V is the restriction mapping of cuts. The mapping ϑAx : lim−→ ΣAx → Ax, of the limit group lim −→ ΣAx of the spectrum σ A x into Ax, defined by relation ϑAx (σ) = s(x), where σ is an arbitrary element of the limit group lim−→ ΣAx and s ∈ σ, is an isomorphism of lim −→ ΣAx onto Ax. 1.2 On the Containing Spaces. In this section we briefly explain the con- struction of the Containing Spaces (see [7], [9]). The spaces of the universal sheaves in the main result of the paper (see below Theorem 1.3.1) will be Containing Spaces. A Containing Space is constructed for a given indexed col- lection S of spaces and it is uniquely determined by a base B for S (in [7] and [9] the base B is called mark and it is denoted by M): B ≡ {{UXδ : δ ∈ τ} : X ∈ S}, © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 151 S .D. Iliadis and Yu. V. Sadovnichy where {UXδ : δ ∈ τ} is an indexed base for the open subsets of X ∈ S, and by a family R of equivalence relations on S: R ≡ {∼t: t ∈ F}. It is required that R satisfies the following conditions: (a) for each t ∈ F the number of equivalence classes of ∼t is finite; (b) for each t1 ⊂ t2 ∈ F, ∼ t2 ⊂ ∼t1; (c) for each X, Y ∈ S the condition X ∼t Y for some t ∈ F, implies that the algebra of subsets of X, generated by the set {UXδ : δ ∈ t}, and the algebra of subsets of Y , generated by the set {UYδ : δ ∈ t}, are isomorphic and the correspondence UXδ → U Y δ , δ ∈ t, generates an isomorphism of these algebras. Such a family is called B-admissible. Also, for each t ∈ F we denote by C(∼t) the set of equivalence classes of ∼t and put C(R) = ∪{C(∼t) : t ∈ F}. The corresponding Containing Space is denoted by T ≡ T(B, R) and its construction is done as follows. Let ∼BR be the equivalence relation on a set of all pairs (x, X), where x ∈ X ∈ S, defined as follows: two such pairs (x, X) and (y, Y ) are ∼BR-equivalent if and only if: (a) X ∼t Y for each t ∈ F and (b) for each δ ∈ τ, x ∈ UXδ if and only if y ∈ UYδ . Then, T is the set of all equivalence classes of ∼ B R and the set BT ≡ {UTδ (H) : δ ∈ τ, H ∈ C(R)}, where UTδ (H) is the set consisting of all points a ∈ T such that there exists an element (x, X) ∈ a for which X ∈ H and x ∈ UXδ , is a base for a topology on T, called standard base (see Corollary 2.8 of [7]). We note that if for some κ ⊂ τ and for each X ∈ S the set {UXδ : δ ∈ κ} is a base for the open subsets of X, then the set {UTδ (H) : δ ∈ κ, H ∈ C(R)} is also a base for the open subsets of the space T (see Corollary 2.8 of [7]). The mapping iXT : X → T, defining by the relation i X T (x) = a ∈ T, where x ∈ X ∈ S and a is the point of T containing the pair (x, X), is an embedding of X into T, which is called natural (see Proposition 2.10 of [7]). In the paper, we shall use also the following notions. Let B1 ≡ {{U X 1,δ : δ ∈ τ} : X ∈ S} and B2 ≡ {{U X 2,δ : δ ∈ τ} : X ∈ S}, where {UX1,δ : δ ∈ τ} and {U X 2,δ : δ ∈ τ} are indexed sets of subsets of X ∈ S (in particular, they may be indexed bases of X) and B2 is a base for S. The base B2 is an extension of B1 if there exists an one-to-one mapping ϑ : τ → τ, called extension mapping, such that UX1,δ = U Y 2,ϑ(δ), δ ∈ τ. We shall also say that for a given X ∈ S, {UX2,δ : δ ∈ τ} is an extension of {U X 1,δ : δ ∈ τ} with the extension mapping ϑ. Let R1 ≡ {∼ t 1: t ∈ F}, and R2 ≡ {∼ t 2: t ∈ F} © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 152 On sheaves of Abelian groups and universality be two families of equivalence relations on S. We say that R2 is a final refine- ment of R1 if for each t ∈ F there exists t ′ ∈ F such that ∼t ′ 2 ⊂ ∼ t 1. A class S of spaces is called saturated if for each indexed collection S of elements of S, there exists a base B0 for S such that for each extension B of B0, there exists a B-admissible family RB of equivalence relations on S with the property that for each admissible family R of equivalence relations on S being a final refinement of RB, the containing space T(B, R) belongs to S (see Section 3 of [7] and Chapter 2 of [9]). The base B0 is called initial base for S (corresponding to the class S) and RB initial family of equivalence relations on S corresponding to B (and the class S). Below, we give some examples of saturated classes of spaces of weight ≤ τ. (1) the class of all T0-spaces (see Propositions 2.9 of [7]); (2) the class of all regular spaces (see Propositions 3.5 of [7]); (3) the class of all completely regular spaces (see Propositions 3.8 of [7]); (4) the class of all spaces of small inductive dimension ind ≤ n ∈ N; (5) the class of all countable-dimensional spaces; (6) the class of all strongly contable-dimensional spaces; (7) the class of all locally finite-dimensional spaces; (8) the intersection of any two saturated classes of spaces. (For the above example (4) see Corrolary 3.1.6 of [9], for (5), (6) and (7) see Proposition 4.4.4 of [9] and for example (8) see Proposition 3.3 of [7]). 1.3 The results. Let S be a class of sheaves. A sheaf p̄ is called proper universal in the class S if p̄ ∈ S and for each p ∈ S there exist a proper isomorphism of p into p̄. The main result of this paper is the following theorem. Theorem 1.3.1. Let Sd and Sr be two saturated classes of spaces of weights ≤ τ. Then, in the class of all sheaves (A, π, X), for which A ∈ Sd and X ∈ Sr, there exists a proper universal element (Ā, π̄, X̄). Since the class of T0-spaces of countable weight and the class of separable metric spaces are saturated classes we have the following corollary. Corollary 1.3.2. In the class of all sheaves (A, π, X), where A is a T0-space of countable weight and X is a separable metrizable space there exists a proper universal element. 2. Proof of the result Lemma 2.1. Let (A, π, X) be a sheaf of Abelian groups. There exists a subset B ⊂ A(sec) such that: (a) ran(B) is a base for the open subsets of A of cardinality w(A) ≤ τ and, therefore, the set dom(B) is a base for the open subsets of X; (b) for each U ∈ dom(B), B(U) is a subgroup of the group A(sec)(U). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 153 S .D. Iliadis and Yu. V. Sadovnichy (c) for each U, V ∈ dom(B) with U ⊂ V the restriction s|U of any cut s ∈ B(V ) belongs to B. Proof. Since A(sec) is a base for A, there exists a subset B0 ⊂ A(sec) such that ran(B0) is a base for the open subsets of A of cardinality w(A). By induction we define the subset Bn ⊂ A(sec), n ∈ N, setting Bn = Bn−1 ∪ (∪{Bn−1+ (U) ⊂ A(sec)(U) : U ∈ dom(B n−1)}) ∪{s|U : s ∈ B n−1(V ), U, V ∈ dom(Bn−1), U ⊂ V }, where Bn−1+ (U) is the subgroup of A(sec)(U) generated by the set B n−1(U). It is easy to see that the set B ≡ ∪{Bn : n ∈ ω} is the required set. � The direct spectrum ΣBx . Let (A, π, X) be a sheaf, B a subset of A(sec), satisfying the conditions of Lemma 2.1, and x ∈ X. By property (a) of this lemma, ran(B) is a base of A and, therefore, the set dom(B) is a base for the open subsets of X. Hence, the set dom(B)(x) is directed by inclusion “ ⊂ ”. By property (c) of Lemma 2.1, for each U, V ∈ dom(B)(x) with U ⊂ V the restriction of rAU,V onto B(V ) is an isomorphism of B(V ) into B(U). We shall denote this restriction by rBU,V . Thus, for each x ∈ X we have a direct spectrum ΣBx of groups: ΣBx ≡ {B(U), r B U,V , dom(B)(x)}. (2.1.1) Let σB be an arbitrary element of the limit group lim −→ ΣBx of the spectrum (2.1.1) and s ∈ σB. We define the mapping ϑBx : lim−→ ΣBx → Ax setting ϑ B x (σ B) = s(x). Lemma 2.2. Let ϑB,Ax : lim−→ ΣBx → lim−→ ΣAx be the mapping defined as follows: for each σB ∈ lim −→ ΣBx we put ϑ B,A x (σ B) = σA, where σA is the element of lim −→ ΣAx containing σ B. Then, ϑB,Ax is well- defined (that is, the element σA is uniquely determined), one-to-one, onto and preserves the group operations, that is it is an isomorphism of lim −→ ΣBx onto lim −→ ΣAx . Moreover, ϑ B x = ϑ A x ◦ ϑ B,A x and, therefore, ϑ B x is an isomorphism and onto mapping. Proof. Since the mappings rBU,V of the spectrum Σ B x are the restrictions of the corresponding mappings rAU,V of the spectrum Σ A x each element σ B of lim −→ ΣBx is contained in an uniquely determined element σA of lim −→ ΣAx , that is the mapping ϑB,Ax is well-defined. We prove that ϑB,Ax is one-to-one. Let σ B 1 and σ B 2 be two distinct elements of lim −→ ΣBx and let σ A 1 , σ A 2 ∈ lim−→ ΣAx such that σ B 1 ⊂ σ A 1 and σ B 2 ⊂ σ A 2 . Suppose that σA1 = σ A 2 and let s1 ∈ σ B 1 and s2 ∈ σ B 2 and, therefore, s1, s2 ∈ σ A 1 . Then, there exists s3 ∈ σ A 1 , which is a restriction of s1 and a restriction of s2. Since dom(B) is a base for the open subsets of X (see property (a) of Lemma 2.1) there exists s0 ∈ B such that x ∈ dom(s0) ⊂ dom(s3) and ran(s0) ⊂ ran(s3). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 154 On sheaves of Abelian groups and universality Then, s0 is the restriction of s3 and, therefore, the restriction of s1 and s2, which contradicts the fact that s1 and s2 belong to distinct elements of lim−→ ΣBx . Thus, ϑB,Ax is one-to-one. We prove that ϑB,Ax is onto. Let σ A be an element of lim −→ ΣAx and s ∈ σ A x . Consider an element s0 ∈ B such that x ∈ dom(s0) ⊂ dom(s) and ran(s0) ⊂ ran(s). Then for the element σB containing s0 we have ϑ B,A x (σ B) = σA, proving that ϑB,Ax is onto. We prove that ϑB,Ax preserves the group operations. Let σ B 1 , σ B 2 ∈ lim−→ ΣBx and let ϑB,Ax (σ B 1 ) = σ A 1 and ϑ B,A x (σ B 2 ) = σ A 2 . Let s1 ∈ σ B 1 and s2 ∈ σ B 2 . Consider an element s0 ∈ B such that dom(s0) ⊂ dom(s1)∩dom(s2). Let s ′ 1 = s1|dom(s0) and s′2 = s2|dom(s0). Then, by proprety (c) of Lemma 2.1, s ′ 1, s ′ 2 ∈ B and by property (b) of this lemma, s′1 + s ′ 2 ∈ B. Therefore, s ′ 1 + s ′ 2 ∈ σ B 1 + σ B 2 . On the other hand, s′1 ∈ σ A 1 and s ′ 2 ∈ σ A 2 and, therefore, s ′ 1 + s ′ 2 ∈ σ A 1 + σ A 2 , proving that ϑB,Ax preserves the sum operation. Similarly, we can prove that ϑB,Ax preserves the taking of the inverse element. Thus, the mapping ϑ B,A x is an isomorphism of lim −→ ΣBx onto lim−→ ΣAx .The relation ϑ B x = ϑ A x ◦ ϑ B,A x is easy to verify. � The indexed collections S, A and X. Consider the saturated classes Sd and Sr of the theorem. By set-theoretical reasons we can suppose that there exists a collection S of sheaves (A, π, X) such that A ∈ Sd, X ∈ Sr and each sheaf (A′, π′, X′), for which A′ ∈ Sd and X ′ ∈ Sr, is isomorphic to an element of S. Moreover, we can suppose that S is indexed by a set Λ: S ≡ {(Aλ, πλ, Xλ) : λ ∈ Λ}. We put A ≡ {Aλ : λ ∈ Λ}, X ≡ {Xλ : λ ∈ Λ}. and consider A and X as indexed by Λ sets of topological spaces. The bases BA and BX for A and X, respectively. For each element (Aλ, πλ, Xλ) ∈ S we consider a subset Bλ ⊂ Aλ(sec) satisfying the conditions of Lemma 2.1. Since |Bλ| = w(Aλ) ≤ τ (see the property (a) of Lemma 2.1), we can suppose that Bλ is indexed by the set τ: Bλ = {s λ η : η ∈ τ}. Furthermore, we put B Aλ 0 ≡ {V Aλ η ≡ ran(s λ η ) : η ∈ τ}, and B Xλ 0 ≡ {V Xλ η ≡ dom(s λ η) : η ∈ τ}. Let θ0 and θ1 be two one-to-one mappings of τ into itself such that |θ0(τ)| = |θ1(τ)|, θ0(τ) ∩ θ1(τ) = ∅ and θ0(τ) ∪ θ1(τ) = τ. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 155 S .D. Iliadis and Yu. V. Sadovnichy (We note that these mappings are independed on λ ∈ Λ.) For each λ ∈ Λ we put W Aλ ζ = V Aλ θ −1 0 (ζ) if ζ ∈ θ0(τ) and W Aλ ζ = π−1 λ (V Xλ θ −1 1 (ζ) ) if ζ ∈ θ1(τ). Therefore, the indexed set BAλ1 ≡ {W Aλ ζ : ζ ∈ τ} is an extension of the indexed base B Aλ 0 of Aλ and, simultaneously, an extension of the indexed set π−1 λ (BXλ0 ) ≡ {π −1 λ (V Xλη ) : η ∈ τ} of subsets of Aλ with the extension mappings θ0 and θ1, respectively. Now, we consider a base B A ≡ {BAλ ≡ {UAλε : ε ∈ τ} : λ ∈ Λ} for A, which is an initial base corresponding to the saturated class Sd and, simultaneously, is an extension of the base B A 1 ≡ {{W Aλ ζ : ζ ∈ τ} : λ ∈ Λ} for A with an extension mapping θA. Also, we consider a base B X ≡ {BXλ ≡ {U Xλ δ : δ ∈ τ} : λ ∈ Λ} for X, which is an initial base corresponding to the saturated class Sr and, simultaneously, is an extension of the base B X 0 ≡ {{V Xλ η : η ∈ τ} : λ ∈ Λ} for X with an extension mapping θX. The families RA and RX of equivalence relations. We denote by RA ≡ {∼ t A : t ∈ F} a BA-admissible family of equivalence relation on A and by RX ≡ {∼ t X : t ∈ F} a BX-admissible family of equivalence relations on X. We suppose that RA and RX satisfy the following conditions: (1) for each λ, µ ∈ Λ and t ∈ F the equivalence Xλ ∼ t X Xµ is true if and only if the equivalence Aλ ∼ t A Aµ is true; (2) for each λ, µ ∈ Λ, t ∈ F, and η1, η2, η ∈ t the equivalence Aλ ∼ t A Aµ implies that the conditions: (21) dom(s λ η1 ) = dom(sλη2) and s λ η1 + sλη2 = s λ η and (22) dom(s µ η1 ) = dom(sµη2) and s µ η1 + sµη2 = s µ η are equivalent; (3) for each λ, µ ∈ Λ, t ∈ F, and η1, η2 ∈ t the equivalence Aλ ∼ t A Aµ implies that the conditions: © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 156 On sheaves of Abelian groups and universality (31) dom(s λ η1 ) = dom(sλη2) and s λ η1 = −sλη2 and (32) dom(s µ η1 ) = dom(sµη2) and s µ η1 = −sµη2 are equivalent ; (4) for each λ, µ ∈ Λ, t ∈ F, and η1, η2 ∈ t the equivalence Aλ ∼ t A Aµ implies that the conditions: (41) ran(s λ η1 ) ⊂ ran(sλη2 ) and (42) ran(s µ η1 ) ⊂ ran(sµη2 ) are equivalent; (5) for each λ, µ ∈ Λ, t ∈ F, and η1, η2 ∈ t the equivalence Aλ ∼ t A Aµ implies that the conditions: (51) dom(s λ η1 ) ⊂ dom(sλη2) and (52) dom(s µ η1 ) ⊂ dom(sµη2) are equivalent. Lemma 2.3. The BA-admissible family RA and the B X-admissible family RX satisfying conditions (1) − (5) exist. Proof. Since the class Sd is saturated there exists a B A-admissible family R0,A ≡ {∼ t 0,A: t ∈ F}, which is initial for the base B A and the class Sd. Similarly, there exists BX-admissible family R0,X ≡ {∼ t 0,X: t ∈ F}, which is initial for the base BX and the class Sr. Let t ∈ F and η1, η2, η ∈ t. We denote by ∼ t i, i ∈ {2, 3, 4, 5}, the equivalence relation on A defined as follows: Aλ ∼ t i Aµ, λ, µ ∈ Λ, if and only if the conditions (i1) and (i2) are equivalent for all indexes η1, η2, η, which belong to t. Obviously, the relations ∼ti, i ∈ {2, 3, 4, 5}, are admissible. Let R1,A ≡ {∼ t 1,A: t ∈ F} be the family of equivalence relations on A, where ∼t1,A=∼ t 0,A ∩(∩{∼ t i: i ∈ {2, 3, 4, 5}}) for each t ∈ F. Now, for each t ∈ F we define the equivalence relation ∼t A on A as follows: Aλ ∼ t A Aµ, λ, µ ∈ Λ, if and only if Aλ ∼ t 1,A Aµ and Xλ ∼ t 0,X Xµ. Also, we define the equivalence relation ∼t X , t ∈ F, on X as follows: Xλ ∼ t X Xµ if and only if Aλ ∼ t A Aµ. It is easy to see that RA ≡ {∼ t A : t ∈ F} and RX ≡ {∼ t X : t ∈ F} are the required families of equivalence relations. � The equivalence relations ∼A and ∼X. We put ∼A= ∩{∼ t A : t ∈ F} and ∼X= ∩{∼ t X : t ∈ F}. The following two lemmas can easily be proved. Lemma 2.4. Let Aλ ∼A Aµ, λ, µ ∈ Λ. Then, the algebra of subsets of Aλ, generated by the set BAλ, and the algebra of subsets of Aµ, generated by the set BAµ, are isomorphic and the correspondence UAλε → U Aµ ε , ε ∈ τ, generates this isomorphism. Therefore, for any κ ⊂ τ the algebra of subsets of Aλ, generated by the set {UAλε : ε ∈ κ}, and the algebra of subsets of Aµ, generated by the © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 157 S .D. Iliadis and Yu. V. Sadovnichy set {U Aµ ε : ε ∈ κ}, are isomorphic and the correspondence U Aλ ε → U Aµ ε , ε ∈ κ, generates this isomorphism. Moreover, for each η, η1, η2 ∈ τ we have: (a) the cut sλη1 is a restriction of the cut s λ η2 if and only if the cut sµη1 is the restriction of the cut sµη2; (b) the equalities dom(sλη1) = dom(s λ η2 ) and sλη = s λ η1 + sλη2 are true if and only if the equalities dom(sµη1) = dom(s µ η2 ) and sµη = s µ η1 + sµη2 are true; (c) the equalities dom(sλη1) = dom(s λ η2 ) and sλη1 = −s λ η2 are true if and only if the equalities dom(sµη1) = dom(s µ η2 ) and sµη1 = −s µ η2 are true. Lemma 2.5. Let Xλ ∼X Xµ, λ, µ ∈ Λ. Then, the algebra of subsets of Xλ, generated by the set BXλ, and the algebra of subsets of Xµ, generated by the set BXµ, are isomorphic and the correspondence U Xλ δ → U Xµ δ , δ ∈ τ, generates this isomorphism. Therefore, for any k ⊂ τ the algebra of subsets of Xλ, generated by the set {U Xλ δ : δ ∈ κ}, and the algebra of subsets of Xµ, generated by the set {U Xµ δ : δ ∈ κ}, are isomorphic and the correspondence U Xλ δ → U Xµ δ , δ ∈ κ, generates this isomorphism. The triad (Ā, π̄, X̄). We put Ā = T(BA, RA), X̄ = T(B X, RX) and define the mapping π̄ as follows. Let a ∈ Ā and (aλ, Aλ) ∈ a for some λ ∈ Λ. Then, we put π̄(a) = x, where x is the point of X̄ containing the pair (πλ(a λ), Xλ). In what follows we shall prove that the triad (Ā, π̄, X̄) is the required universal sheaf. Lemma 2.6. The mapping π̄ is correctly defined (that is, it is independent from the element (aλ, Aλ) ∈ a considered in its definition). Proof. Let a ∈ Ā and (aλ, Aλ), (b µ, Aµ) ∈ a, that is (a λ, Aλ) and (b µ, Aµ) are ∼B A RA -equivalent. We must prove that if πλ(a λ) = xλ and πµ(b µ) = yµ, then (xλ, Xλ) and (y µ, Xµ) are ∼ B X RX -equivalent, that is Xλ ∼X Xµ and for each δ ∈ τ either xλ ∈ UXλ δ and yµ ∈ U Xµ δ or xλ /∈ UXλ δ and yµ /∈ U Xµ δ . Since (aλ, Aλ) and (bµ, Aµ) are ∼ B A RA -equivalent, Aλ ∼A Aµ. By the condition (1) of the definitions of RA and RX we have Xλ ∼X Xµ. Suppose that there exists δ0 ∈ τ such that, for example, x λ ∈ UXλ δ0 and yµ /∈ U Xµ δ0 . Then, aλ ∈ π−1 λ (UXλ δ0 ) and bµ /∈ π−1µ (U Xµ δ0 ). Since the set B Xλ 0 is a base for the open subsets of Xλ, there exists η ∈ τ such that xλ ∈ V Xλη ⊂ U Xλ δ0 . Let δ1 = θX(η) and, therefore, V Xλ η = U Xλ δ1 and V Xµ η = U Xµ δ1 . Then, U Xλ δ1 ⊂ U Xλ δ0 . By Lemma 2.5 we have U Xµ δ1 ⊂ U Xµ δ0 and, therefore, yµ /∈ U Xµ δ1 = V Xµ η . Since B A is an extension of π−1 λ (BXλ0 ) with the extension mapping θA ◦ θ1 we have U Aν ε = π −1 ν (V Xν η ) for each ν ∈ Λ and ε = θA(θ1(η)). Therefore, for λ and µ we have U Aλ ε = π −1 λ (V Xλη ) and U Aµ ε = π −1 µ (V Xµ η ), respectively, and hence a λ ∈ UAλε and b µ 6= U Aµ ε , which contradicts the fact that (aλ, Aλ) and (b µ, Aµ) are ∼ B A RA -equivalent. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 158 On sheaves of Abelian groups and universality The following lemma can easily be verified. Lemma 2.7. For each λ ∈ Λ the following relation is true: π̄ ◦ iAλ Ā = iXλ X̄ ◦ πλ, (2.7.1) where i Aλ Ā is the natural embedding of Aλ into Ā and i Xλ X̄ is the natural em- bedding of Xλ into X̄. Lemma 2.8. For each η ∈ τ and H ∈ C(RA) we have π̄(UĀε (H)) = U X̄ δ (L), where ε = θA(θ0(η)), δ = θX(η) and L is the element of C(RX), consisting of all elements Xλ ∈ X for which Aλ ∈ H (we shall say that L and H correspond each other). Proof. Let η ∈ τ, ε = θA(θ0(η)), δ = θX(η) and H ∈ C(RA). Then, for each λ ∈ Λ, UAλε = im(s λ η). By the definition of the elements of the standard base of the Containing Spaces, we have UĀε (H) = ∪{i Aλ Ā (UAλε ) : Aλ ∈ H} = ∪{i Aλ Ā (ran(sλη )) : Aλ ∈ H}. Therefore, using relation (2.7.1), we have π̄(UĀε (H)) = ∪{π̄(i Aλ Ā (ran(sλη ))) : Aλ ∈ H} = ∪{i Xλ X̄ (πλ(ran(s λ η))) : Xλ ∈ L} = ∪{i Xλ X̄ (dom(sλη)) : Xλ ∈ L} = ∪{iXλ X̄ (UXλ δ )) : Xλ ∈ L} = U X̄ δ (L). � Proposition 2.9. The mapping π̄ is continuous. Proof. Since the set {UX̄δ (L) : δ ∈ θX(τ), L ∈ C(RX)} is a base of the space X̄ it suffices to prove that the set π̄−1(ŪXδ (L)) is open in Ā for each δ ∈ θX(τ) and L ∈ C(RX). Let δ be a fixed element of θX(τ) and L a fixed element of C(RX). Let η = θ−1 X (δ) and ε = θA(θ1(η)). Then, for each ν ∈ Λ we have π−1ν (U Xν δ ) = UAνε . (2.9.1) We shall prove the following equality, which will prove the continuity of π̄: π̄−1(UX̄δ (L)) = U Ā ε (H), (2.9.2) where H is the element of C(RA) corresponding to L. Let a ∈ π̄ −1(UX̄δ (L)), that is π̄(a) ≡ x ∈ UX̄δ (L). Let (x λ, Xλ) ∈ x and (a µ, Aµ) ∈ a. Since π̄(a) = x, by the definition of π̄ we have (πµ(a µ), Xµ) ∈ x. This means that Xλ ∼X Xµ, that is Xµ ∈ L and, therefore, Aµ ∈ H. Also, πµ(a µ) ∈ U Xµ δ and, therefore, aµ ∈ π−1µ (U Xµ δ ) = UAµε (2.9.3) © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 159 S .D. Iliadis and Yu. V. Sadovnichy (see relation (2.9.1)). Since Aµ ∈ H, the relation (2.9.3) shows that a ∈ U Ā ε (H), proving that the left side of the relation (2.9.2) is contained in the right. Conversly, let a ∈ UĀε (H) and (a µ, Aµ) ∈ a. Then, Aµ ∈ H and aµ ∈ U Aµ ε . Therefore, Xµ ∈ L and a µ ∈ π−1µ (U Xµ δ ) (see relation 2.9.1) or πµ(a µ) ∈ U Xµ δ . This means that π̄(a) ∈ UX̄δ (L) and, therefore, a ∈ π̄ −1(UX̄δ (L)), proving that the right side of (2.9.2) is contained in the left, completing the proof of the proposition. � The set Āx, x ∈ X̄. For each x ∈ X̄ we put Āx = {a ∈ Ā : π̄(a) = x}. We shall prove that Āx is an Abelian group. First we shall prove the following lemma. Lemma 2.10. Let (Aλ, πλ, Xλ) and (Aµ, πµ, Xµ) be two elements of S such that Aλ ∼A Aµ and let x ∈ X̄. Then, for each two elements (xλ, Xλ), (x µ, Xµ) ∈ x ∈ X̄ there exists an isomorphism ϑxλxµ of Aλ,xλ onto Aµ,xµ such that: (a) ϑxλxλ is the identical isomorphism; (b) ϑ xµ xν ◦ ϑ xλ xµ = ϑxλxν , where (x ν, Xν) ∈ x. Moreover, for each aλ ∈ Aλ,xλ, we have (aλ, Aλ) ∼ B A RA (ϑx λ xµ(a λ), Aµ). (2.10.1) Proof. Let (xλ, Xλ), (x µ, Xµ) ∈ x ∈ X̄ for some fixed λ, µ ∈ Λ. Then, x λ ∈ UXλ δ for some δ ∈ θX(τ) if and only if x µ ∈ U Xµ δ , that is xλ ∈ dom(sλη) for some η ∈ τ, if and only if xµ ∈ dom(sµη ). Denote by κ all such η. By Lemma 2.5 it follows that the mapping dom(sλη) → dom(s µ η ), η ∈ κ, is an isomorphism of the directed by inclusion set dom(Bλ)(x λ) onto the directed by inclusion set dom(Bµ)(x µ). Let σλ ≡ {sλη : η ∈ τ(σ λ)} be an element of the limit group lim −→ Σ Bλ xλ , where τ(σλ) is the set of all η ∈ τ for which sλη ∈ σ λ. Consider the set σµ(λ) ≡ {sµη : η ∈ τ(σ λ)} of sections of Aµ. Let s λ η1 , sλη2 ∈ σ λ such that ran(sλη1) ⊂ ran(s λ η2 ), that is the section sλη1 is the restriction of the section s λ η2 . By relation Aλ ∼A Aµ and Lemma 2.4 it follows that ran(sµη1 ) ⊂ ran(s µ η2 ), that is the section sµη1 is the restriction of the section sµη2. This means that the set σ µ(λ) is a subset of an unique determined element σµ of the limit group lim −→ Σ Bµ xµ . Similarly, the constructed element σµ of the limit group lim −→ Σ Bµ xµ defines a set σλ(µ) of sections of Aλ. By construction, σ λ ⊂ σλ(µ) and, therefore, © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 160 On sheaves of Abelian groups and universality σλ = σλ(µ). Similarly, σµ = σµ(λ). We will say that σλ ∈ lim −→ ΣBλ xλ and σµ ∈ lim −→ Σ Bµ xµ correspond each other. Thus, we have defined mutually inverse, one-to-one and onto mappings ϑ(xλ, xµ) : lim −→ Σ Bλ xλ → lim −→ Σ Bµ xµ and ϑ(x µ, xλ) : lim −→ Σ Bµ xµ → lim−→ Σ Bλ xλ . We prove that the mapping ϑ(xλ, xµ) is an isomorphism, that is it preserves the group operations. Let σλ1 , σ λ 2 ∈ lim−→ ΣBλ xλ and let sλη1 ∈ σ λ 1 and s λ η2 ∈ σλ2 . Since the set dom(Bλ)(x λ) is a base of Xλ at the point xλ, there exists sλη ∈ Bλ such that U ≡ dom(sλη) ⊂ dom(s λ η1 ) ∩ dom(sλη2). By condition (c) of Lemma 2.1 the restrictions sλη1|U and s λ η2 |U belong to Bλ and by condition (b), s λ η1 |U + s λ η2 |U belongs to Bλ. Then, there exist η ′ 1, η ′ 2, η ′ 0 ∈ τ such that sλη′ 1 = sλη1|U ∈ σ λ 1 , s λ η′ 2 = sλη2|U ∈ σ λ 2 and s λ η′ 0 = sλη′ 1 + sλη′ 2 ∈ σλ1 + σ λ 2 . Let ϑ(xλ, xµ)(σλ1 ) = σ µ 1 , ϑ(x λ, xµ)(σλ2 ) = σ µ 2 , and ϑ(x λ, xµ)(σλ) = σµ. By Lemmas 2.4 and 2.5 we have V ≡ dom(sµη ) ⊂ dom(s µ η1 ) ∩ dom(s µ η2), s µ η′ 1 = sµη1|V ∈ σ µ 1 , s µ η′ 2 = sµη2|V ∈ σ µ 2 , and, s µ η′ 0 = s µ η′ 1 + s µ η′ 2 ∈ σ µ 1 + σ µ 2 Therefore, ϑ(xλ, xµ)(σλ1 + σ λ 2 ) = ϑ(x λ, xµ)(σλ1 ) + ϑ(x λ, xµ)(σλ2 ). Similarly, we prove that ϑ(xλ, xµ)(−σλ1 ) = −ϑ(x λ, xµ)(σλ1 ). Thus, the mapping ϑ(xλ, xµ) and, therefore, the mapping ϑ(xµ, xλ) is an isomorphism and onto. The required isomorphism ϑx λ xµ of Aλ,xλ onto Aµ,xµ is defined by setting ϑx λ xµ = ϑ Bµ xµ ◦ ϑ(x λ, xµ) ◦ (ϑ Bλ xλ )−1. Conditions (a) and (b) of the lemma can easily be verified. We prove relation (2.10.1). Let aλ ∈ Aλ,xλ and σ λ 0 = (ϑ Bλ xλ )−1(aλ). Then, ϑx λ xµ(a λ) = ϑ Bµ xµ (ϑ(x λ, xµ)(σλ0 )) = ϑ Bµ xµ (σ µ 0 ), where σ µ 0 is the element lim−→ Σ Bµ xµ corresponding to σ λ 0 . Thus, it suffices to prove that the pairs (ϑBλ xλ (σλ0 ), Aλ) and (ϑ B µ xµ (σ µ 0 ), Aµ) belong to the same element of Ā. Let UAλε be an element of B Aλ containing ϑ Bλ xλ (σλ). We need to prove that ϑ Bµ xµ (σ µ) ∈ UAµε . (2.10.2) Since the set {ran(s) : s ∈ σλ} is a base for the open subsets of Aλ at the point ϑ Bµ xµ (σ µ), there exists η ∈ τ such that sλη ∈ σ λ and ϑBλ xλ (σλ) ∈ ran(sλη) ⊂ U Aλ ε . © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 161 S .D. Iliadis and Yu. V. Sadovnichy Since σµ corresponds to σλ, sµη ∈ σ. By the definition of the mapping ϑ Bµ xµ , ϑ Bµ xµ (σ µ) = sµη(x µ) ∈ ran(sµη ). Since Aλ ∼A Aµ, the condition ran(s λ η) ⊂ U Aλ ε implies that ran(sµη ) ⊂ U Aµ ε , proving relation (2.10.2). � Proposition 2.11. For each x ∈ X̄ the set Āx is an Abelian group and for each (xλ, Xλ) ∈ x, λ ∈ Λ, the natural embedding i Aλ Ā of Aλ into Ā maps the fiber Aλ,xλ of Aλ onto the set Āx. Proof. Let x ∈ X̄, (xλ, Xλ) ∈ x for some fixed λ ∈ Λ and a ∈ Āx. By relation (2.7.1) it follows that iAλ Ā (Aλ,xλ) ⊂ Āx. We must prove that i Aλ Ā (Aλ,xλ) = Āx. By the definition of the Containing Spaces there exists µ ∈ Λ and a point aµ ∈ Aµ such that (a µ, Aµ) ∈ a. Using relation (2.7.1) we can see that (xµ, Xµ) ∈ x, where x µ = πµ(a µ). Let aλ = (ϑx λ xµ) −1(aµ). Then, by the relation (2.10.1), (aλ, Aλ) ∼ B A RA (aµ, Aµ) and, therefore, (a λ, Aλ) ∈ a, that is i Aλ Ā (aλ) = a, proving that i Aλ Ā maps the fiber Aλ,xλ of Aλ onto the set Āx. Now, on the set Āx, x ∈ X̄, we define the group operations. Let a1, a2 ∈ Āx and let (aλ1 , Aλ) ∈ a1 and (a λ 2 , Aλ) ∈ a2. Then, we put a1 + a2 = a, where a is the element of Āx, containing the pair (a λ 1 + a λ 2 , Aλ). Also, we consider that −a1 is the element of Āx, containing the pair (−a λ 1, Aλ). Obviously, by these operations Āx becomes an Abelian group such that the restriction onto Aλ,xλ of the natural embedding i Aλ Ā of Aλ into Ā, is an isomorphism of Aλ,xλ onto Āx. It remains to prove that the defined operations are independent of the element (xλ, Xλ) ∈ x. Let (x ν, Xν) ∈ x for some ν ∈ Λ, (a ν 1, Aν) ∈ a1 and (a ν 2, Aν) ∈ a2. We need to prove that the pair (a λ 1 + a λ 2, Aλ) and (a ν 1 + a ν 2, Aν) belong to the same element of Āx. Since, by Lemma 2.10, ϑ x λ xν is an isomorphism we have ϑx λ xν (a λ 1 + a λ 2 ) = ϑ x λ xν (a λ 1 ) + ϑ x λ xν (a λ 2 ). On the other hand, since the pairs (aλ1 , Aλ) and (ϑ x λ xν (a λ 1 ), Aν) belong to the same element of Ā we have (ϑx λ xν (a λ 1), Aν) ∈ a1 and since (a ν 1, Aν) ∈ a1 we have ϑx λ xν (a λ 1) = a ν 1. Similarly, ϑ x λ xν (a λ 2 ) = a ν 2. Therefore, ϑ x λ xν (a λ 1 + a λ 2) = a ν 1 + a ν 2. Since, by Lemma 2.10, the pairs (aλ1 + a λ 2, Aλ) and (ϑ x λ xν (a λ 1 + a λ 2 ), Aν) belong to the same element of Ā, the pairs (aλ1 +a λ 2 , Aλ) and (a ν 1 +a ν 2, Aν) also belong to the same element of Ā, proving that the sum operation is independent of the element (xλ, Xλ) ∈ x. Similarly, we prove that the operation of taking the inverse element is independent of (xλ, Xλ) ∈ x. The proof of the proposition is completed. � The mappings ̟Ā and iĀ. We put Ā ⊠ Ā ≡ {(a, b) ∈ Ā × Ā : π̄(a) = π̄(b)} © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 162 On sheaves of Abelian groups and universality and define the mappings ̟Ā : Ā ⊠ Ā → Ā and iĀ : Ā → Ā setting ̟Ā(a, b) = a + b and iĀ(a) = −a. Proposition 2.12. The mappings ̟Ā and iĀ are continuous. Proof. We prove that ̟Ā is continuous. Let (a1, a2) ∈ Ā ⊠ Ā and a ≡ a1 + a2. Let U be an open neighbourhood of a in Ā. We must find open neighbourhoods U1 and U2 of a1 and a2, respectively, in Ā such that ̟Ā((U1 × U2) ∩ (Ā ⊠ Ā)) ⊂ U. (2.12.1) Without loss of generality, we can suppose that U is an element UĀε (H) of the standard base of Ā, where ε is a fixed element of τ and H is a fixed element of C(∼t A ) for some fixed t ∈ F. Moreover, we can suppose that ε ∈ θA(θ0(τ)). This means that for each Aµ ∈ H we have U Aµ ε = ran(s µ η ), where η = (θA ◦ θ0) −1(ε) and, therefore, UĀε (H) = ∪{i Aµ Ā (UAµε ) : Aµ ∈ H} = ∪{i Aµ Ā (ran(sµη )) : Aµ ∈ H}. Let λ be a fixed element of Λ such that Aλ ∈ H and a ∈ i Aλ Ā (ran(sλη )). Therefore, there exists a point aλ ∈ ran(sλη ) ⊂ Aλ such that i Aλ Ā (aλ) = a, that is (aλ, Aλ) ∈ a. Let πλ(a λ) = xλ ∈ Xλ. Then, by relation (2.7.1), iXλ X̄ (xλ) = π̄(a) ≡ x ∈ X̄ and, therefore, (xλ, Xλ) ∈ x. By Lemma 2.10, there are points aλ1 , a λ 2 ∈ Aλ,xλ such that (a λ 1 , Aλ) ∈ a1, (a λ 2, Aλ) ∈ a2 and (aλ1 + a λ 2, Aλ) ∈ a. Since (a λ, Aλ) ∈ a we have a λ 1 + a λ 2 = a λ ∈ ran(sλη). Since the mapping ̟λ is continuous, there exist ε1, ε2 ∈ τ such that a λ 1 ∈ U Aλ ε1 , aλ2 ∈ U Aλ ε2 and ̟λ((U Aλ ε1 × UAλε2 ) ∩ (Aλ ⊠ Aλ)) ⊂ ran(s λ η). (2.12.2) Without loss of generality, we can suppose that ε1, ε2 ∈ θA(θ0(τ)), that is there exist cuts sλη1 and s λ η2 such that ran(sλη1 ) = U Aλ ε1 , ran(sλη2) = U Aλ ε2 , dom(sλη1) = dom(sλη2) ⊂ dom(s λ η). Condition (b) of Lemma 2.1 implies that there exists η0 ∈ τ such that s λ η0 = sλη1 + s λ η2 and, therefore, dom(sλη0 ) = dom(s λ ηi ), i = 1, 2. In this case, the left side of the relation (2.12.2) takes the form ̟λ((ran(s λ η1 ) × ran(sλη2 )) ∩ (Aλ ⊠ Aλ)). (2.12.3) Since Aλ ⊠ Aλ = ∪{Aλ,xλ × Aλ,xλ : x λ ∈ Xλ}, the expression (2.12.3) takes the form ̟λ(∪{(ran(s λ η1 ) ∩ Aλ,xλ) × (ran(s λ η2 ) ∩ Aλ,xλ) : x λ ∈ Xλ}). (2.12.4) Since ran(sληi ) ∩ Aλ,xλ = ∅ if x λ /∈ dom(sληi), i = 1, 2, (2.12.5) © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 163 S .D. Iliadis and Yu. V. Sadovnichy and ran(sληi) ∩ Aλ,xλ = {s λ ηi (xλ)} if xλ ∈ dom(sληi ), i = 1, 2, (2.12.6) the expression (2.12.4) takes the form ̟λ(∪{{s λ η1 (xλ)} × {sλη2(x λ)} : xλ ∈ dom(sλη0 )}) = ̟λ(∪{{(s λ η1 (xλ), sλη2(x λ))} : xλ ∈ dom(sλη0)}) = ∪{{sλη1(x λ) + sλη2(x λ)} : xλ ∈ dom(sλη0)} = ∪{{(sλη1 + s λ η2 )(xλ)} : xλ ∈ dom(sλη0 )}) = ran(s λ η0 ). Thus, relation (2.12.2) implies that ran(sλη0) ⊂ ran(s λ η). Let t′ ∈ F and {η1, η2, η, η0} ∪ t ⊂ t ′. Denote by H′ the element of C(∼t ′ A ) containing Aλ. Therefore, H ′ ⊂ H. Then, for each Aµ ∈ H ′ we have Aµ ∼ t ′ A Aλ, ran(s µ η1 ) = UAµε1 , ran(s µ η2 ) = UAµε2 . (2.12.7) By condition (4) and (5) of the definitions of the families RA and RX, we have dom(sµη1) = dom(s µ η2 ) = dom(sµη0 ), (2.12.8) ran(sµη1 + s µ η2 ) = ran(sµη0 ) ⊂ ran(s µ η ). (2.12.9) We shall prove that the sets UĀεi (H ′) = ∪{i Aµ Ā (ran(sµηi )) : Aµ ∈ H ′}, i = 1, 2, are the required open neighbourhoods Ui of ai. Obviously, ai ∈ Ui. By Lemma 2.11 we have Ā∆Ā = ∪{Āx × Āx : x ∈ X̄} = {i Aν Ā (Aν,xν ) × i Aν Ā (Aµ,xν ) : Aν ∈ A, x ν ∈ Xν} Using this relation and the relation (2.12.7) we have ̟Ā((U Ā ε1 (H′) × UĀε2(H ′)) ∩ (Ā ⊠ Ā)) = ̟Ā(∪{(i Aµ Ā (ran(sµη1 )) ∩ i Aν Ā (Aν,xν )) × (i Aξ Ā (ran(sξη2)) ∩ i Aν Ā (Aν,xν )) : : Aξ ∈ H ′, Aν ∈ A, x ν ∈ X̄})). (2.12.10) If Aν /∈ H ′, then the intersections in the right side of the above equality are empty. Therefore, we can suppose that Aν ∈ H ′. In this case, relations (2.12.7)- (2.12.9) are true if we replace the letter “ µ ” by “ ν ”. Let a1 ∈ i Aµ Ā (ran(sµη1 ))∩ iAν Ā (Aν,xν ). Then relation a1 ∈ i Aµ Ā (ran(sµη1 )) implies that there exists a µ 1 ∈ ran(sµη1) and (a µ 1 , Aµ) ∈ a1 and the relation a1 ∈ i Aν Ā (Aν,xν ) implies that there exists aν1 ∈ Aν,xν such that (a ν 1, Aν) ∈ a1 and πν(a ν 1) = x ν. From these it follows that Aµ ∼A Aν and since a µ 1 ∈ U Aµ ε1 = ran(s µ η1 ) we have aν1 ∈ UAνε1 = ran(s ν η1 ). Therefore, aν1 = s ν η1 (xν). Similarly, if a2 ∈ i Aξ Ā (ran(sξη2 )) ∩ iAν Ā (Aν,xν ), then there exists a ν 2 ∈ ran(s ν η2 ) and πν(a ν 2) = x ν. Thus, aν1, a ν 2 ∈ Aν,xν . Then, the equality (2.12.10) can be continued as follows: ̟Ā(∪{(i Aν Ā (sνη1(x ν)), iAν Ā (sνη2(x ν))) : Aν ∈ H ′, xν ∈ dom(sνη0 )}) = © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 164 On sheaves of Abelian groups and universality ∪{iAν Ā (sνη1(x ν) + sνη2(x ν)) : Aν ∈ H ′, xν ∈ dom(sνη0)} = ∪{iAν Ā ((sνη1 + s ν η2 )(xν)) : Aν ∈ H ′, xν ∈ dom(sνη0 )} = ∪{iAν Ā (ran(sνη0 )) : Aν ∈ H ′} ⊂ ∪{iAν Ā (ran(sνη)) : Aν ∈ H ′} = UĀε (H ′) ⊂ UĀε (H). We note that for the first equality of the above expression we use the fact that the restriction of the mapping iAν Ā on the set Aν,xν is an isomorphism. Thus, we proved relation (2.12.1), which means that the mapping ̟Ā is con- tinuous. Similarly, we prove that the mapping iĀ is continuous. The proof of the proposition is completed. � Proposition 2.13. For each point a ∈ Ā there exists an open neighbourhood U of a in Ā such that π̄ maps U homeomorphically onto an open set of X̄. Proof. Let a ∈ Ā and (aλ, Aλ) for some λ ∈ Λ. There exists an open neigh- bourhood of aλ in Aλ, which πλ maps homeomorphically onto an open subset of Xλ. Since B Aλ 0 is a base for the open subsets of Aλ, without loss of gener- ality, we can suppose that this open neighbourhood is an element V Aλη of this base. Let t ∈ F such that η ∈ t and H be the element of C(∼t A ) containing Aλ. We prove that U Ā ε (H), where ε = θA(θ0(η)), is the required open subset U. By Lemma 2.8 we have π̄(UĀε (H)) = U X̄ δ (L), where δ = θX(η) and L is the element of C(RX) corresponding to H. First, we prove that the restriction of π̄ onto the open subset UĀε (H) of Ā is one- to-one. Indeed, if not, there are two distinct points b1 and b2 of U Ā ε (H) such that π̄(b1) = π̄(b2) ≡ x ∈ X̄, that is b1, b2 ∈ Āx. Let (x ν, Xν) ∈ x for some ν ∈ Λ. By Proposition 2.11 there exist points bν1, b ν 2 ∈ Aν,xν such that iAν Ā (bν1) = b1 and i Aν Ā (bν2) = b2, that is (b ν 1, Aν) ∈ b1 and (b ν 2, Aν) ∈ b2. We have bν1, b ν 2 ∈ U Aν ε = V Aν η = ran(s ν η), b ν 1 6= b ν 2 and πν(b ν 1) = πν(b ν 2) = x ν, which contradicts the fact that the πν maps homeomorphically the set ran(s ν η) onto an open subset of Xν. Since the restriction of the mapping π̄ onto the set UĀε (H) is one-to-one we can consider the inverse mapping, denoted by s̄ε,H, of the set U X̄ δ (L) onto U Ā ε (H). We shall prove that s̄ε,H is continuous. Let x1 ∈ U X̄ δ (L) and a1 = s̄ε,H(x1). Let also U1 be an arbitrary open neighbourhood of a1 in U Ā ε (H). Since U Ā ε (H) is open in Ā, without loss of generality, we can suppose that U1 belongs to the standard base of Ā, that is it has the form UĀε1(H1). Moreover, we can suppose that ε1 = θA(θ0(η1)) for some η1 ∈ τ and H1 ∈ C(∼ t1 A ), where t1 is an element of F such that t ⊂ t1. Thus, we have a1 ∈ U Ā ε1 (H1) ⊂ U Ā ε (H). By Lemma 2.8, π̄(UĀε1(H1)) = U X̄ δ1 (L1), © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 165 S .D. Iliadis and Yu. V. Sadovnichy where δ1 = θX(η1) and L1 is the element of C(RX) corresponding to H1. As the above, the restriction of the mapping π̄ onto UĀε1(H1) is one-to-one and, therefore, we can consider the inverse mapping, denoted by s̄ε1,H1, of the set UX̄δ1 (L1) onto the set U Ā ε1 (H1). The mapping s̄ε1,H1 coincides with the restriction of the mapping s̄ε,H onto the set U X̄ δ1 (L1), that is s̄ε,H(U X̄ δ1 (L1)) = UĀε1(H1), which shows that s̄ε,H is continuous. Thus, π̄ maps the set U Ā ε (H) homeomorphically onto an open subset of X̄. The proof of the proposition is completed. � The final of the proof of Theorem 1.3.1 Relation (2.7.1) implies that for each (Aλ, πλ, Xλ) ∈ S the natural embedding i Xλ X̄ of Xλ into X̄ is the induced mapping of the natural embedding i Aλ Ā of Aλ into Ā. Proposition 2.11 shows that the embedding iAλ Ā is proper. 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