@ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 2, 2011 pp. 193-211 Universal elements for some classes of spaces D. N. Georgiou, S. D. Iliadis and A. C. Megaritis∗ Abstract In the paper [4] two dimensions, denoted by dm and Dm, are defined in the class of all Hausdorff spaces. The dimension Dm does not have the universality property in the class of separable metrizable spaces because the family of all such spaces X with Dm(X) ≤ 0 coincides with the family of all totally disconnected spaces in which there are no universal elements (see [5]). In [3] we gave the dimension-like functions dm IK,IB IE and Dm IK,IB IE , where IK is a class of subsets, IE a class of spaces and IB a class of bases and we proved that in the families IP(dm IK,IB IE ≤ κ) and IP(Dm IK,IB IE ≤ κ) of all spaces X for which dm IK,IB IE (X) ≤ κ and Dm IK,IB IE (X) ≤ κ, respectively there exist universal elements. In this paper, we give some new dimension-like functions and define using these definitions classes of spaces in which there are universal elements. 2010 MSC: 54B99, 54C25. Keywords: Dimension-like function, saturated class, universal space. 1. Introduction and preliminaries Agreement. All spaces are assumed to be T0-spaces of weight ≤ τ, where τ is a fixed infinite cardinal. The set of all finite subsets of τ is denoted by F and the first infinite cardinal is denoted by ω. The cardinality of a set X is denoted by |X|. The class of all ordinals is denoted by O. We also consider two symbols: −1 and ∞. It is assumed that −1 < α < ∞ for every α ∈ O. In the proof of the main results of this paper widely we use notions and nota- tions from [2]. For this reason we start given some of them. ∗Work supported by the Caratheodory Programme of the University of Patras. 194 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis We shall use the symbol “ ≡ ” in order to introduce new notations without mention this fact. If “ ∼ ” is an equivalence relation on a non-empty set X, then the set of all equivalence classes of ∼ is denoted by C(∼). Let S be an indexed collection of spaces. An indexed collection M ≡ {{UXδ : δ ∈ τ} : X ∈ S} (1) where {UXδ : δ ∈ τ} is an indexed base for X, is called a co-mark of S. The co-mark M of S is said to be a co-extension of a co-mark M + ≡ {{V Xδ : δ ∈ τ} : X ∈ S} of S if there exists a one-to-one mapping θ of τ into itself such that for every X ∈ S and for every δ ∈ τ, V Xδ = U X θ(δ). The corresponding mapping θ is called an indicial mapping from M+ to M. Let R1 ≡ {∼ s 1: s ∈ F} and R0 ≡ {∼ s 0: s ∈ F} be two indexed families of equivalence relations on S. It is said that R1 is a final refinement of R0 if for every s ∈ F there exists t ∈ F such that ∼ t 1⊆∼ s 0. An indexed family R ≡ {∼s: s ∈ F} of equivalence relations on S is said to be admissible if the following conditions are satisfied: (a) ∼∅= S × S, (b) for every s ∈ F the number of ∼s-equivalence classes is finite, and (c) ∼s ⊆ ∼t, if t ⊆ s. We denote by C(R) the set ∪{C(∼s) : s ∈ F}. The minimal ring of subsets of S containing C(R) is denoted by C♦(R). Consider the co-mark (1) of S. We denote by RM ≡ {∼ s M: s ∈ F} the indexed family of equivalence relations ∼sM on S defined as follows: for every X, Y ∈ S we set X ∼sM Y if and only if there exists an isomorphism i of the algebra of subsets of X generated by the set {UXδ : δ ∈ s} onto the algebra of subsets of Y generated by the set {UYδ : δ ∈ s} such that i(U X δ ) = U Y δ , for every δ ∈ s. Also, we set ∼∅ M = S × S. An admissible family R of equivalence relations on S is said to be M-admissible if R is a final refinement of RM. Let R ≡ {∼s: s ∈ F} be an M-admissible family of equivalence relations on S. On the set of all pairs (x, X), where X ∈ S and x ∈ X, we consider an equivalence relation, denoted by ∼MR , as follows: (x, X) ∼ M R (y, Y ) if and only if X ∼s Y for every s ∈ F, and either x ∈ UXδ and y ∈ U Y δ or x /∈ U X δ and y /∈ UYδ for every δ ∈ τ. The set of all equivalence classes of the relation ∼ M R is denoted by T(M, R) or simply by T. For every H ∈ C♦(R) the set of all a ∈ T(M, R) for which there exists an element (x, X) ∈ a such that X ∈ H is denoted by T(H). For every δ ∈ τ and H ∈ C♦(R) we denote by UTδ (H) the set of all a ∈ T(M, R) for which there exists an element (x, X) ∈ a such that X ∈ H and x ∈ UXδ . Universal elements for some classes of spaces 195 For every subset κ of τ and L ∈ C♦(R) we set (1) BT ♦ ≡ {UTδ (H) : δ ∈ τ and H ∈ C ♦(R)}. (2) BT ♦,κ ≡ {U T δ (H) : δ ∈ κ and H ∈ C ♦(R)}. (3) BL ♦,κ ≡ {U T δ (H) ∈ B T ♦,κ : H ⊆ L}. Under some simple (set-theoretical) conditions on R the set BT♦ is a base for a topology on the set T(M, R) such that the corresponding space is a T0-space of weight ≤ τ. Moreover, if for every X ∈ S the set {UXδ : δ ∈ κ} is a base for X, then the set BT♦,κ is a base for the same topology on T(M, R). Therefore, the family BL♦,κ is a base for T(L). (See Corollary 1.2.8 and Proposition 1.2.9 in [2]). For every element X of S there exists a natural embedding iXT of X into the space T(M, R) defined as follows: for every x ∈ X, iXT (x) = a, where a is the element of T(M, R) containing the pair (x, X). Thus, we have constructed containing space T(M, R) for S of weight ≤ τ. Suppose that for every X ∈ S a subset QX of X is given. The set Q ≡ {QX : X ∈ S} (2) is called a restriction of S. Let IF be a class of subsets. A restriction Q of an indexed collection S of spaces is said to be a IF-restriction if (QX, X) ∈ IF for every X ∈ S. Consider the restriction (2) of S. The trace on Q of the co-mark M of S is the co-mark M|Q ≡ {{U X δ ∩ Q X : δ ∈ τ} : QX ∈ Q} of Q. The trace on Q of an equivalence relation ∼ on S is the equivalence relation on Q denoted by ∼|Q and defined as follows: Q X∼|QQ Y if and only if X ∼ Y . Let R ≡ {∼s: s ∈ F} be an indexed family of equivalence relations on S. The trace on Q of the family R is the family R|Q ≡ {∼ s|Q : s ∈ F} of equivalence relations on Q. The trace on Q of an element H of C♦(R) is the element H|Q ≡ {Q X ∈ Q : X ∈ H} of C♦(R|Q). The M-admissible family R of equivalence relations on S is said to be (M, Q)- admissible if R|Q is an M|Q-admissible family of equivalence relations on Q. If R is an (M, Q)-admissible family of equivalence relations on S, then we can consider the containing space T(M|Q, R|Q) for the indexed collection Q corresponding to the co-mark M|Q and the M|Q-admissible family R|Q. The containing space T(M|Q, R|Q) is denoted briefly by T|Q. There exists a natural embedding of T(M|Q, R|Q) into T(M, R). So we can consider the containing space T(M|Q, R|Q) as a subspace of the space T(M, R). The subsets of this form will be called specific subsets of T(M, R). A class IP of spaces is said to be saturated if for every indexed collection S of spaces belonging to IP there exists a co-mark M+ of S satisfying the following 196 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis condition: for every co-extension M of M+ there exists an M-admissible family R+ of equivalence relations on S such that for every admissible family R of equivalence relations on S, which is a final refinement of R+, and for every L ∈ C♦(R) the space T(L) belongs to IP. The co-mark M+ is said to be an initial co-mark of S corresponding to the class IP and the family R is said to be an initial family of S corresponding to the co-mark M and the class IP. Agreement. In what follows we denote by ν a fixed cardinal greater than ω and less than or equal to τ. Notation. For every dimension-like function dfν, with as domain the class of all spaces and as range the class O ∪ {−1, ∞}; and for every α ∈ {−1} ∪ O, we denote by IP(dfν ≤ α) the class of all spaces X with dfν(X) ≤ α. 2. The dimension-like functions: dm IK,IB IE,ν and Dm IK,IB IE,ν In this section we give some new dimension-like functions and define using these definitions classes of spaces in which there are universal elements. The proofs of these results are similar to the proofs of the results in [3], for this reason are omitted. Definition 2.1 (see [1]). Let A and B be two disjoint subsets of a space X. We say that a subset L of X separates A and B if there exist two open subsets U and W of X such that: (a) A ⊆ U, B ⊆ W , (b) U ∩ W = ∅, and (c) X \ L = U ∪ W . Definition 2.2 (see [3]). A class IE of spaces is said to be IB-hereditary- separated, where IB is a class of bases, if for every element X of IE there exists a IB-base BX ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1 and Uδ2 of B X with Cl(Uδ1) ∩ Cl(Uδ2) = ∅ there exists a subset L of X separating the sets Cl(Uδ1) and Cl(Uδ2) and belonging to IE. We note that if IE is IB-hereditary-separated, then ∅ ∈ IE. This follows by the fact that the empty set is the unique subset of X separating the elements ∅ and X of BX. Definition 2.3. Let IB be a class of bases, IE a IB-hereditary-separated class of spaces, and IK a class of subsets with (X, X) ∈ IK for every space X. We denote by dm IK,IB IE,ν and Dm IK,IB IE,ν the dimension-like functions with as domain the class of all spaces and as range the class O ∪ {−1, ∞} satisfying the following conditions: (1) dm IK,IB IE,ν (X) = Dm IK,IB IE,ν (X) = −1 if and only if X ∈ IE. (2) Dm IK,IB IE,ν (X) ≤ α, α ∈ O, if and only if there exists a IB-base B X ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1, Uδ2 of B X with Cl(Uδ1) ∩ Cl(Uδ2) = ∅ there exists a subset L of X separating Cl(Uδ1) and Cl(Uδ2) with dm IK,IB IE,ν (L) < α. Universal elements for some classes of spaces 197 (3) dm IK,IB IE,ν (X) ≤ α, α ∈ O, if and only if X = ∪{Si : i ∈ ν} such that: (a) the subset Si of X is closed, (b) (Si, X) ∈ IK, and (c) Dm IK,IB IE,ν (Si) ≤ α, i ∈ ν. Therefore, dm IK,IB IE,ν (X) = ∞ (respectively, Dm IK,IB IE,ν (X) = ∞) if and only if the inequality dm IK,IB IE,ν (X) ≤ α (respectively, Dm IK,IB IE,ν (X) ≤ α) is not true for every α ∈ O. Remark 2.4. (1) In order that the above definition to be well defined we need to show that if for a space X we have Dm IK,IB IE,ν (X) = dm IK,IB IE,ν (X) = −1, then Dm IK,IB IE,ν (X) ≤ 0 and dm IK,IB IE,ν (X) ≤ 0. For dimension-like function Dm IK,IB IE,ν this follows immediately by the fact that X ∈ IE and the class IE is IB-hereditary-separated. For dimension-like function dm IK,IB IE,ν , we have (a) X = {Si : i ∈ ν}, where Si = X, (b) (X, X) ∈ IK, and (c) Dm IK,IB IE,ν (X) = −1 ≤ 0, which means that dm IK,IB IE,ν (X) ≤ 0. (2) For ν = ω the dimension-like functions dm IK,IB IE,ν and Dm IK,IB IE,ν coincide with the dimension-like functions dm IK,IB IE and Dm IK,IB IE , respectively which are defined in [3]. Proposition 2.5. For every space X we have dm IK,IB IE,ν (X) ≤ Dm IK,IB IE,ν (X). Proposition 2.6. For every space X, Dm IK,IB IE,ν (X) ∈ {−1, ∞}∪τ + and, there- fore, dm IK,IB IE,ν (X) ∈ {−1, ∞} ∪ τ +. Theorem 2.7. Let IB be a saturated class of bases, IE a saturated IB-hereditary- separated class of spaces, and IK a saturated class of subsets with (X, X) ∈ IK for every space X. Then, for every κ ∈ {−1} ∪ ω the classes IP(dm IK,IB IE,ν ≤ κ) and IP(Dm IK,IB IE,ν ≤ κ) are saturated. Corollary 2.8. For every κ ∈ ω in the classes IP(dm IK,IB IE,ν ≤ κ) and IP(Dm IK,IB IE,ν ≤ κ) there exist universal elements. Corollary 2.9. Let IP be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and 198 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis (e) the class of all (completely) regular spaces X of weight ≤ τ such that ind(X) ≤ α ∈ τ+. Then, for every κ ∈ ω in the classes IP(dm IK,IB IE,ν ≤ κ) ∩ IP and IP(Dm IK,IB IE,ν ≤ κ) ∩ IP there exist universal elements. 3. The dimension-like functions: w-dm IK,IB IE,ν and w-Dm IK,IB IE,ν Definition 3.1. A class IE of spaces is said to be IB-weakly-hereditary-separated, where IB is a class of bases, if for every element X of IE there exists a IB-base BX ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1 and Uδ2 of B X with Cl(Uδ1)∩Uδ2 = ∅ there exists a subset L of X separating the sets Cl(Uδ1) and Uδ2 and belonging to IE. We note that if IE is IB-weakly-hereditary-separated, then ∅ ∈ IE. This follows by the fact that the empty set is the unique subset of X separating the elements ∅ and X of BX. Definition 3.2. Let IB be a class of bases, IE a IB-weakly-hereditary-separated class of spaces, and IK a class of subsets with (X, X) ∈ IK for every space X. We denote by w-dm IK,IB IE,ν and w-Dm IK,IB IE,ν the dimension-like functions with as domain the class of all spaces and as range the class O ∪ {−1, ∞} satisfying the following conditions: (1) w-dm IK,IB IE,ν (X)=w-Dm IK,IB IE,ν (X) = −1 if and only if X ∈ IE. (2) w-Dm IK,IB IE,ν (X) ≤ α, where α ∈ O, if and only if there exists a IB-base BX ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1, Uδ2 of BX with Cl(Uδ1) ∩ Uδ2 = ∅ there exists a subset L of X separating Cl(Uδ1) and Uδ2 with w-dm IK,IB IE,ν (L) < α. (3) w-dm IK,IB IE,ν (X) ≤ α, α ∈ O, if and only if X = ∪{Si : i ∈ ν} such that: (a) the subset Si of X is closed, (b) (Si, X) ∈ IK, and (c) w- Dm IK,IB IE,ν (Si) ≤ α, i ∈ ν. Therefore, w-dm IK,IB IE,ν (X) = ∞ (respectively, w-Dm IK,IB IE,ν (X) = ∞) if and only if the inequality w-dm IK,IB IE,ν (X) ≤ α (respectively, w-Dm IK,IB IE,ν (X) ≤ α) is not true for every α ∈ O. Remark 3.3. In order that the above definition to be well defined we need to show that if for a space X we have w-Dm IK,IB IE,ν (X)=w-dm IK,IB IE,ν (X) = −1, then w-Dm IK,IB IE,ν (X) ≤ 0 and w-dm IK,IB IE,ν (X) ≤ 0. For dimension-like function w-Dm IK,IB IE,ν this follows immediately by the fact that X ∈ IE and the class IE is IB-weakly-hereditary-separated. For dimension-like function w-dm IK,IB IE,ν , we have (a) X = {Si : i ∈ ν}, where Si = X, (b) (X, X) ∈ IK, and (c) w-Dm IK,IB IE,ν (X) = −1 ≤ 0, which means that w-dm IK,IB IE,ν (X) ≤ 0. Universal elements for some classes of spaces 199 Proposition 3.4. For every space X we have w-dm IK,IB IE,ν (X) ≤ w-Dm IK,IB IE,ν (X). (3) Proof. Let w-Dm IK,IB IE,ν (X) = α ∈ {−1, ∞} ∪ O. The inequality (3) is clear if α = −1 or α = ∞. Suppose that α ∈ O. We have X = ∪{Si : i ∈ ν}, where Si = X. Since (Si, X) = (X, X) ∈ IK and w-Dm IK,IB IE,ν (Si) = w-Dm IK,IB IE,ν (X) ≤ α, the condition (3) of Definition 3.2 implies that w-dm IK,IB IE,ν (X) ≤ α. � Proposition 3.5. For every space X, w-Dm IK,IB IE,ν (X) ∈ {−1, ∞} ∪ τ +, and, therefore w-dm IK,IB IE,ν (X) ∈ {−1, ∞} ∪ τ +. Proof. Suppose that the proposition is not true. Let α be the minimal element of O \ τ+ such that there exists a space X with w-Dm IK,IB IE,ν (X) = α. Let BX = {Uδ : δ ∈ τ} be the IB-base for X mentioned in condition (2) of Definition 3.2. Denote by P the set of all pairs (δ1, δ2) ∈ τ × τ with Cl(Uδ1) ∩ Uδ2 = ∅. For every (δ1, δ2) ∈ P let L(δ1, δ2) be a subset of X separating the sets Cl(Uδ1) and Uδ2 with w-dm IK,IB IE,ν (L(δ1, δ2)) = β(δ1, δ2) < α. First we suppose that β(δ1, δ2) < τ + for every (δ1, δ2) ∈ P . Since |P | ≤ τ there exists an ordinal β ∈ τ+ such that β(δ1, δ2) < β for every (δ1, δ2) ∈ P . Then, w-dm IK,IB IE,ν (L(δ1, δ2)) < β and, by condition (2) of Definition 3.2, w-Dm IK,IB IE,ν (X) ≤ β, which is a contradiction. Now, we suppose that there exists (δ1, δ2) ∈ P such that τ + ≤ β(δ1, δ2). Since w-dm IK,IB IE,ν (L(δ1, δ2)) = β(δ1, δ2), there exist closed subsets S L(δ1,δ2) i of L(δ1, δ2), i ∈ ν, such that: (a) L(δ1, δ2) = ∪{S L(δ1,δ2) i : i ∈ ν}, (b) (S L(δ1,δ2) i , L(δ1, δ2)) ∈ IK, and (c) w-Dm IK,IB IE,ν (S L(δ1,δ2) i ) = βi ≤ β(δ1, δ2) < α. If βi < τ + for all i ∈ ν, then there exists an ordinal β ∈ τ+ such that βi ≤ β, which means that w-Dm IK,IB IE,ν (S L(δ1,δ2) i ) ≤ β. Therefore, w-Dm IK,IB IE,ν (L(δ1, δ2)) ≤ β < τ + ≤ β(δ1, δ2), which is a contradiction. Thus, there exists i ∈ ν such that τ+ ≤ w-Dm IK,IB IE,ν (S L(δ1,δ2) i ) < α. The last relation contradicts the choice of the ordinal α completing the proof of the proposition. � 200 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis Theorem 3.6. Let IB be a saturated class of bases, IE a saturated IB-weakly- hereditary-separated class of spaces, and IK a saturated class of subsets such that (X, X) ∈ IK for every space X. Then, for every κ ∈ {−1} ∪ ω the classes IP(w-dm IK,IB IE,ν ≤ κ) and IP(w-Dm IK,IB IE,ν ≤ κ) are saturated. Proof. We prove the theorem by induction on κ. Let κ = −1. Then, a space X belongs to IP(w-Dm IK,IB IE,ν ≤ −1) if and only if X belongs to IE, that is IP(w-Dm IK,IB IE,ν ≤ −1) = IE. Therefore, IP(w-Dm IK,IB IE,ν ≤ −1) is a saturated class of spaces. Similarly, the class IP(w-dm IK,IB IE,ν ≤ −1) is saturated. Let κ ∈ ω. Suppose that the classes IP(w-dm IK,IB IE,ν ≤ m) and IP(w-Dm IK,IB IE,ν ≤ m) are saturated, m ∈ {−1}∪κ. We prove that the classes IP(w-Dm IK,IB IE,ν ≤ κ) and IP(w-dm IK,IB IE,ν ≤ κ) are also saturated. First we prove that IP(w-Dm IK,IB IE,ν ≤ κ) is a saturated class. Let S be an indexed collection of elements of IP(w-Dm IK,IB IE,ν ≤ κ). For every X ∈ S let BX ≡ {V Xε : ε ∈ τ} be an indexed IB-base for X satisfying condition (2) of Definition 3.2. Then, there exist (a) an indexed set {LXη : η ∈ τ} of subsets of X, (b) two indexed sets {W Xη : η ∈ τ} and {O X η : η ∈ τ} of open subsets of X, and (c) a one-to-one mapping ϕ of τ × τ onto τ such that (1) For every ε1, ε2 ∈ τ and η = ϕ(ε1, ε2) we have (d) Cl(V Xε1 ) ⊆ W X η , V X ε2 ⊆ OXη , (e) W Xη ∩ O X η = ∅, and (f) X \ LXη = W X η ∪ O X η , in the case, where Cl(V Xε1 )∩V X ε2 = ∅, and LXη = ∅ in the case, where Cl(V X ε1 )∩ V Xε2 6= ∅. (2) For every η ∈ τ, w-dm IK,IB IE,ν (L X η ) ≤ κ − 1. For every η ∈ τ we set Lη = {L X η : X ∈ S}, Wη = {W X η : X ∈ S}, and Oη = {O X η : X ∈ S}. By the above property (2), Lη is an indexed collection of elements of the class IPκ−1 ≡ IP(w-dm IK,IB IE,ν ≤ κ − 1). By inductive assumption the class IPκ−1 is saturated. Therefore, there exists an initial co-mark M+ Lη of Lη corresponding to the class IPκ−1. Denote by Mη a co-mark of S such that its trace on Lη is Universal elements for some classes of spaces 201 a co-extension of the co-mark M+ Lη . The existence of such a co-mark is easily proved. Consider the co-indication N ≡ {{V Xε : ε ∈ τ} : X ∈ S} of the IB-co-base B ≡ {BX : X ∈ S} of S. Since IB is a saturated class of bases there exists an initial co-mark M+IB of S corresponding to the co-indication N of B and the class IB. In particular, M+IB is a co-extension of N. By Lemma 2.1.2 of [2], there exists a co-mark M+ of S, which a co-extension of the co-marks M+IB and Mη for every η ∈ τ. In particular, M + is a co-extension of N. We show that M+ is an initial co-mark of S corresponding to the class IP(w-Dm IK,IB IE,ν ≤ κ). Indeed, let M ≡ {{UXδ : δ ∈ τ} : X ∈ S} be an arbitrary co-extension of M+. Then, M is a co-extension of the co-marks M + IB, N, and Mη for every η ∈ τ. Denote by ϑ an indicial mapping from N to M. Then, for every X ∈ IE, V Xε = U X ϑ(ε), ε ∈ τ. Obviously, the co-mark M|Lη is a co-extension of the co-mark M+ Lη of Lη. Let R+IB be an initial family of equivalence relations on S corresponding to the co-mark M, the co-indication N of B, and the class IB. Let also R+ Lη be an initial family of equivalence relations on Lη corresponding to the co-mark M|Lη and the class IPκ−1. Denote by Rη the family of equivalence relations on S such that the trace on Lη of Rη is the family R + Lη . By Lemma 2.1.1 of [2], there exists an admissible family R+ of equivalence relations on S, which is a final refinement of the families R+IB and Rη for every η ∈ τ. In particular, R+ is M-admissible. Without loss of generality, we can suppose that R+ is (M, Wη)-admissible, (M, Oη)-admissible, (M, Co(Wη))- admissible, and (M, Co(Oη))-admissible. We prove that R + is an initial family of S corresponding to the co-mark M of S and the class IP(w-Dm IK,IB IE,ν ≤ κ). For this purpose we consider an arbitrary admissible family R of equivalence relations on S, which is a final refinement of R+, and prove that for every L ∈ C♦(R) the space T(L) belongs to IP(w-Dm IK,IB IE,ν ≤ κ). Let L ∈ C ♦(R). Since IB is a saturated class, we have (BL ♦,ϑ(τ) , T(L)) ∈ IB. We show that the base BL ♦,ϑ(τ) of T(L) satisfies condition (2) of Definition 3.2, that is for every UTδ1(H1) and U T δ2 (H2) of B L ♦,ϑ(τ) (where H1, H2 ⊆ L), with ClT(L)(U T δ1 (H1)) ∩ U T δ2 (H2) = ∅ (4) there exists a subset L of T(L) separating ClT(L)(U T δ1 (H1)) and U T δ2 (H2) such that w-dm IK,IB IE,ν (L) ≤ κ − 1. Consider two elements UTδ1(H1), U T δ2 (H2) of B L ♦,ϑ(τ) satisfying relation (4). First we suppose that H1 ∩ H2 = ∅. Then, 202 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis (g) ClT(L)(U T δ1 (H1)) ⊆ T(H1), U T δ2 (H2) ⊆ T(L \ H1), (h) T(H1) ∩ T(L \ H1) = ∅, and (i) T(L) = T(H1) ∪ T(L \ H1). Therefore, the empty set separates the sets ClT(L)(U T δ1 (H1)) and U T δ2 (H2). Since w-dm IK,IB IE,ν (∅) = −1 < κ, we have w-Dm IK,IB IE,ν (T(L)) ≤ κ. Now, we suppose that H1 ∩ H2 6= ∅. Let H = H1 ∩ H2, ϑ −1(δ1) = ε1, ϑ−1(δ2) = ε2, and η = ϕ(ε1, ε2). We prove that T(H|Lη) separates the sets ClT(L)(U T δ1 (H1)) and U T δ2 (H2), and w-dm IK,IB IE,ν (T(H|Lη )) ≤ κ − 1 < κ. Since IPκ−1 is a saturated class of spaces, the subspace T(H|Lη ) of T(M|Lη , R|Lη) belongs to IPκ−1. Hence, w-dm IK,IB IE,ν (T(H|Lη )) ≤ κ − 1 < κ. We prove that the subset T(H|Lη ) of T(L) separates ClT(L)(U T δ1 (H1)) and UTδ2(H2). Suppose that X ∈ H. Since the subsets Cl(V X ε1 ) and V Xε2 of X are disjoint, by condition (1) we have (k) Cl(V Xε1 ) ⊆ W X η , V X ε2 ⊆ OXη , (l) W Xη ∩ O X η = ∅, and (m) X \ LXη = W X η ∪ O X η . The above relations imply that (n) ClT(L)(U T δ1 (H)) ⊆ T(H|Wη ) = T|Wη ∩ T(H), UTδ2(H) ⊆ T(H|Oη ) = T|Oη ∩ T(H), (o) T(H|Wη ) ∩ T(H|Oη ) = ∅, and (p) T(H) \ T(H|Lη) = T(H|Wη ) ∪ T(H|Oη ). Since the restriction Wη of S is open and the family R is (M, Co(Wη))- admissible, by Lemma 1.4.7 of [2], the subset T|Wη of T is open. Similarly, the subset T|Oη of T is open. Also, since the subset T(H) of T is open and T(H) ⊆ T(L), the sets T(H|Wη) and T(H|Oη ) are open in T(L). Setting W = T(H1 \ H) ∪ T(H|Wn) and O = T(L \ H1) ∪ T(H|On) we have (q) ClT(L)(U T δ1 (H1)) ⊆ W , U T δ2 (H2) ⊆ O, (r) W ∩ O = ∅, and (s) T(L) \ T(H|Lη ) = W ∪ O. Therefore, the subset T(H|Lη ) of T(L) separates the sets ClT(L)(U T δ1 (H1)) and UTδ1(H2). Thus, the class IP(w − Dm IK,IB IE,ν ≤ κ) is saturated. Now, we prove that the class IP(w-dm IK,IB IE,ν ≤ κ) is saturated. Let S be a indexed collection of elements of IP(w-dm IK,IB IE,ν ≤ κ). For every X ∈ S there exists an indexed set {QXi : i ∈ ν} of subsets of X such that Universal elements for some classes of spaces 203 (3) X = ∪{QXi : i ∈ ν}. (4) For every i ∈ ν, the subset QXi of X is closed and (Q X i , X) ∈ IK. (5) For every i ∈ ν, w-Dm IK,IB IE,ν (Q X i ) ≤ κ. We set Qi = {Q X i : X ∈ S}, i ∈ ν. By the preceding, the class IP ≡ IP(w-Dm IK,IB IE,ν ≤ κ) is saturated. By property (5), Qi is an indexed collec- tion of elements of the class IP. Therefore, there exists an initial co-mark M+ Qi of Qi corresponding to the class IP. Denote by Mi a co-mark of S such that its trace on Qi is a co-extension of the co-mark M + Qi . By property (4), the restriction Qi of S is a IK-restriction. Since IK is a saturated class of subsets, for every i ∈ ν there exists an initial co-mark M+IK,i of S corresponding to the IK-restriction Qi. By Lemma 2.1.2 of [2], there exists a co-mark M+ of S, which a co-extension of the co-marks Mi and M + IK,i for every i ∈ ν. We show that M + is an initial co-mark of S corresponding to the class IP(w-dm IK,IB IE,ν ≤ κ). Indeed, let M ≡ {{UXδ : δ ∈ τ} : X ∈ S} be an arbitrary co-extension of M+. Then, M is a co-extension of the co-marks Mi and M + IK,i and the co-mark M|Qi is a co-extension of the co-mark M + Qi of Qi, i ∈ ν. Let R+ Qi be an initial family of equivalence relations on Qi corresponding to the co-mark M|Qi and the class IP. Denote by Ri the family of equivalence relations on S such that the trace on Qi of Ri is the family R + Qi . Let also R+IK,i be an initial family of equivalence relations on S corresponding to the co-mark M and the IK-restriction Qi. By Lemma 2.1.1 of [2], there exists an admissible family R+ of equivalence relations on S, which is a final refinement of the families Ri and R + IK,i, i ∈ ν. Therefore, R+ is an M-admissible family. We prove that R+ is an initial family of S corresponding to the co-mark M of S and the class IP(w-dm IK,IB IE,ν ≤ κ). For this purpose, we consider an arbitrary admissible family R of equivalence relations on S, which is a final refinement of R+. Then, R is a final refinement of the families Ri and R + IK,i for every i ∈ ν. We need to prove that for every L ∈ C♦(R), T(L) ∈ IP(w-dm IK,IB IE,ν ≤ κ). Let L ∈ C♦(R). It suffices to show that T(L) = ∪{Ti(L) : i ∈ ν} such that (t) the subset Ti(L) of T(L) is closed, (u) (Ti(L), T(L)) ∈ IK, and (v) w-Dm IK,IB IE,ν (Ti(L)) ≤ κ, i ∈ ν. We set Ti(L) = T(L|Qi ), i ∈ ν. It is easy to verify that the subset T(L|Qi ) of T(L) is closed and T(L) = ∪{T(L|Qi) : i ∈ ν}. Since IK is a saturated class 204 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis of subsets, (T(L|Qi ), T(L)) ∈ IK. Since IP is a saturated class, the subspace T(L|Qi) of T(M|Qi, R|Qi) belongs to IP. Hence, w-Dm IK,IB IE,ν (T(L|Qi )) ≤ κ. Thus, by condition (3) of Definition 3.2, w-dm IK,IB IE,ν (T(L)) ≤ κ proving that the class IP(w-dm IK,IB IE,ν ≤ κ) is saturated. � Corollary 3.7. For every κ ∈ ω in the classes IP(w-dm IK,IB IE,ν ≤ κ) and IP(w-Dm IK,IB IE,ν ≤ κ) there exist universal elements. Corollary 3.8. Let IP be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and (e) the class of all (completely) regular spaces X of weight ≤ τ such that ind(X) ≤ α ∈ τ+. Then, for every κ ∈ ω in the classes IP(w-dm IK,IB IE,ν ≤ κ) ∩ IP and IP(w-Dm IK,IB IE,ν ≤ κ) ∩ IP there exist universal elements. 4. The dimension-like functions: s-dm IK,IB IE,ν and s-Dm IK,IB IE,ν Definition 4.1. A class IE of spaces is said to be IB-strong-hereditary-separated, where IB is a class of bases, if for every element X of IE there exists a IB-base BX ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1 and Uδ2 of B X with Uδ1 ∩ Uδ2 = ∅ there exists a subset L of X separating the sets Uδ1 and Uδ2 and belonging to IE. We note that if IE is IB-strong-hereditary-separated, then ∅ ∈ IE. This follows by the fact that the empty set is the unique subset of X separating the elements ∅ and X of BX. Definition 4.2. Let IB be a class of bases, IE a IB-strong-hereditary-separated class of spaces, and IK a class of subsets with (X, X) ∈ IK for every space X. We denote by s-dm IK,IB IE,ν and s-Dm IK,IB IE,ν the dimension-like functions with as domain the class of all spaces and as range the class O ∪ {−1, ∞} satisfying the following conditions: (1) s-dm IK,IB IE,ν (X)=s-Dm IK,IB IE,ν (X) = −1 if and only if X ∈ IE. (2) s-Dm IK,IB IE,ν (X) ≤ α, where α ∈ O, if and only if there exists a IB-base BX ≡ {Uδ : δ ∈ τ} for X such that for every two elements Uδ1, Uδ2 of Universal elements for some classes of spaces 205 BX with Uδ1 ∩ Uδ2 = ∅ there exists a subset L of X separating Uδ1 and Uδ2 with s-dm IK,IB IE,ν (L) < α. (3) s-dm IK,IB IE,ν (X) ≤ α, α ∈ O, if and only if X = ∪{Si : i ∈ ν} such that: (a) the subset Si of X is closed, (b) (Si, X) ∈ IK, and (c) s- Dm IK,IB IE,ν (Si) ≤ α, i ∈ ν. Therefore, s-dm IK,IB IE,ν (X) = ∞ (respectively, s-Dm IK,IB IE,ν (X) = ∞) if and only if the inequality s-dm IK,IB IE,ν (X) ≤ α (respectively, s-Dm IK,IB IE,ν (X) ≤ α) is not true for every α ∈ O. Remark 4.3. In order that the above definition to be well defined we need to show that if for a space X we have s-Dm IK,IB IE,ν (X)=s-dm IK,IB IE,ν (X) = −1, then s-Dm IK,IB IE,ν (X) ≤ 0 and s-dm IK,IB IE,ν (X) ≤ 0. For dimension-like function s-Dm IK,IB IE,ν this follows immediately by the fact that X ∈ IE and the class IE is IB-strong-hereditary-separated. For dimension-like function s-dm IK,IB IE,ν , we have (a) X = {Si : i ∈ ν}, where Si = X, (b) (X, X) ∈ IK, and (c) s-Dm IK,IB IE,ν (X) = −1 ≤ 0, which means that s-dm IK,IB IE,ν (X) ≤ 0. Proposition 4.4. For every space X we have s-dm IK,IB IE,ν (X) ≤ s-Dm IK,IB IE,ν (X). (5) Proof. Let s-Dm IK,IB IE,ν (X) = α ∈ {−1, ∞} ∪ O. The inequality (5) is clear if α = −1 or α = ∞. Suppose that α ∈ O. We have X = ∪{Si : i ∈ ν}, where Si = X. Since (Si, X) = (X, X) ∈ IK and s-Dm IK,IB IE,ν (Si) = s-Dm IK,IB IE,ν (X) ≤ α, the condition (3) of Definition 4.2 implies that s-dm IK,IB IE,ν (X) ≤ α. � Proposition 4.5. For every space X, s-Dm IK,IB IE,ν (X) ∈ {−1, ∞} ∪ τ +, and, therefore s-dm IK,IB IE,ν (X) ∈ {−1, ∞} ∪ τ +. Proof. Suppose that the proposition is not true. Let α be the minimal element of O \ τ+ such that there exists a space X with s-Dm IK,IB IE,ν (X) = α. Let BX = {Uδ : δ ∈ τ} be the IB-base for X mentioned in condition (2) of Definition 4.2. Denote by P the set of all pairs (δ1, δ2) ∈ τ × τ with Uδ1 ∩ Uδ2 = ∅. For every (δ1, δ2) ∈ P let L(δ1, δ2) be a subset of X separating the sets Uδ1 and Uδ2 with s-dm IK,IB IE,ν (L(δ1, δ2)) = β(δ1, δ2) < α. First we suppose that β(δ1, δ2) < τ + for every (δ1, δ2) ∈ P . Since |P | ≤ τ there exists an ordinal β ∈ τ+ such that β(δ1, δ2) < β for every (δ1, δ2) ∈ P . Then, 206 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis s-dm IK,IB IE,ν (L(δ1, δ2)) < β and, by condition (2) of Definition 4.2, s-Dm IK,IB IE,ν ≤ β, which is a contradiction. Now, we suppose that there exists (δ1, δ2) ∈ P such that τ + ≤ β(δ1, δ2). Since s-dm IK,IB IE,ν (L(δ1, δ2)) = β(δ1, δ2), there exist closed subsets S L(δ1,δ2) i of L(δ1, δ2), i ∈ ν, such that: (a) L(δ1, δ2) = ∪{S L(δ1,δ2) i : i ∈ ν}, (b) (S L(δ1,δ2) i , L(δ1, δ2)) ∈ IK, and (c) s-Dm IK,IB IE,ν (S L(δ1,δ2) i ) = βi ≤ β(δ1, δ2) < α. If βi < τ + for all i ∈ ν, then there exists an ordinal β ∈ τ+ such that βi ≤ β, which means that s-Dm IK,IB IE,ν (S L(δ1,δ2) i ) ≤ β. Therefore, s-dm IK,IB IE,ν (L(δ1, δ2)) ≤ β < τ + ≤ β(δ1, δ2), which is a contradiction. Thus, there exists i ∈ ν such that τ+ ≤ s-Dm IK,IB IE,ν (S L(δ1,δ2) i ) < α. The last relation contradicts the choice of the ordinal α completing the proof of the proposition. � Theorem 4.6. Let IB be a saturated class of bases, IE a saturated IB-strong- hereditary-separated class of spaces, and IK a saturated class of subsets such that (X, X) ∈ IK for every space X. Then, for every κ ∈ {−1} ∪ ω the classes IP(s-dm IK,IB IE,ν ≤ κ) and IP(s-Dm IK,IB IE,ν ≤ κ) are saturated. Proof. We prove the theorem by induction on κ. Let κ = −1. Then, a space X belongs to IP(s-Dm IK,IB IE,ν ≤ −1) if and only if X belongs to IE, that is IP(s-Dm IK,IB IE,ν ≤ −1) = IE. Therefore, IP(s-Dm IK,IB IE,ν ≤ −1) is a saturated class of spaces. Similarly, the class IP(s-dm IK,IB IE,ν ≤ −1) is saturated. Let κ ∈ ω. Suppose that the classes IP(s-dm IK,IB IE,ν ≤ m) and IP(s-Dm IK,IB IE,ν ≤ m) are saturated, m ∈ {−1} ∪ κ. We prove that the classes IP(s-Dm IK,IB IE,ν ≤ κ) and IP(s-dm IK,IB IE,ν ≤ κ) are also saturated. First we prove that IP(s-Dm IK,IB IE,ν ≤ κ) is a saturated class. Let S be an indexed collection of elements of IP(s-Dm IK,IB IE,ν ≤ κ). For every X ∈ S let BX ≡ {V Xε : ε ∈ τ} be an indexed IB-base for X satisfying condition (2) of Definition 4.2. Then, there exist (a) an indexed set {LXη : η ∈ τ} of subsets of X, (b) two indexed sets {W Xη : η ∈ τ} and {O X η : η ∈ τ} of open subsets of X, and (c) a one-to-one mapping ϕ of τ × τ onto τ such that Universal elements for some classes of spaces 207 (1) For every ε1, ε2 ∈ τ and η = ϕ(ε1, ε2) we have (d) V Xε1 ⊆ W X η , V X ε2 ⊆ OXη , (e) W Xη ∩ O X η = ∅, and (f) X \ LXη = W X η ∪ O X η , in the case, where V Xε1 ∩V X ε2 = ∅, and LXη = ∅ in the case, where V X ε1 ∩V Xε2 6= ∅. (2) For every η ∈ τ, s-dm IK,IB IE,ν (L X η ) ≤ κ − 1. For every η ∈ τ we set Lη = {L X η : X ∈ S}, Wη = {W X η : X ∈ S}, and Oη = {O X η : X ∈ S}. By the above property (2), Lη is an indexed collection of elements of the class IPκ−1 ≡ IP(s-dm IK,IB IE,ν ≤ κ − 1). By inductive assumption the class IPκ−1 is saturated. Therefore, there exists an initial co-mark M+ Lη of Lη corresponding to the class IPκ−1. Denote by Mη a co-mark of S such that its trace on Lη is a co-extension of the co-mark M+ Lη . The existence of such a co-mark is easily proved. Consider the co-indication N ≡ {{V Xε : ε ∈ τ} : X ∈ S} of the IB-co-base B ≡ {BX : X ∈ S} of S. Since IB is a saturated class of bases there exists an initial co-mark M+IB of S corresponding to the co-indication N of B and the class IB. In particular, M+IB is a co-extension of N. By Lemma 2.1.2 of [2], there exists a co-mark M+ of S, which a co-extension of the co-marks M+IB and Mη for every η ∈ τ. In particular, M + is a co-extension of N. We show that M+ is an initial co-mark of S corresponding to the class IP(s-Dm IK,IB IE,ν ≤ κ). Indeed, let M ≡ {{UXδ : δ ∈ τ} : X ∈ S} be an arbitrary co-extension of M+. Then, M is a co-extension of the co-marks M + IB, N, and Mη for every η ∈ τ. Denote by ϑ an indicial mapping from N to M. Then, for every X ∈ IE, V Xε = U X ϑ(ε), ε ∈ τ. Obviously, the co-mark M|Lη is a co-extension of the co-mark M+ Lη of Lη. Let R+IB be an initial family of equivalence relations on S corresponding to the co-mark M, the co-indication N of B, and the class IB. Let also R+ Lη be an initial family of equivalence relations on Lη corresponding to the co-mark M|Lη and the class IPκ−1. Denote by Rη the family of equivalence relations on S such that the trace on Lη of Rη is the family R + Lη . By Lemma 2.1.1 of [2], there exists an admissible family R+ of equivalence relations on S, which is a final refinement of the families R+IB and Rη for every 208 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis η ∈ τ. In particular, R+ is M-admissible. Without loss of generality, we can suppose that R+ is (M, Wη)-admissible, (M, Oη)-admissible, (M, Co(Wη))- admissible, and (M, Co(Oη))-admissible. We prove that R + is an initial family of S corresponding to the co-mark M of S and the class IP(s-Dm IK,IB IE,ν ≤ κ). For this purpose we consider an arbitrary admissible family R of equivalence relations on S, which is a final refinement of R+, and prove that for every L ∈ C♦(R) the space T(L) belongs to IP(s-Dm IK,IB IE,ν ≤ κ). Let L ∈ C ♦(R). Since IB is a saturated class, we have (BL ♦,ϑ(τ) , T(L)) ∈ IB. We show that the base BL ♦,ϑ(τ) of T(L) satisfies condition (2) of Definition 4.2, that is for every UTδ1(H1) and U T δ2 (H2) of B L ♦,ϑ(τ) (where H1, H2 ⊆ L), with UTδ1(H1) ∩ U T δ2 (H2) = ∅ (5) there exists a subset L of T(L) separating UTδ1(H1) and U T δ2 (H2) such that s-dm IK,IB IE,ν (L) ≤ κ − 1. Consider two elements UTδ1(H1), U T δ2 (H2) of B L ♦,ϑ(τ) satisfying relation (5). First we suppose that H1 ∩ H2 = ∅. Then, (g) UTδ1(H1) ⊆ T(H1), U T δ2 (H2) ⊆ T(L \ H1), (h) T(H1) ∩ T(L \ H1) = ∅, and (i) T(L) = T(H1) ∪ T(L \ H1). Therefore, the empty set separates the sets UTδ1(H1) and U T δ2 (H2). Since s-dm IK,IB IE,ν (∅) = −1 < κ, we have s-Dm IK,IB IE,ν (T(L)) ≤ κ. Now, we suppose that H1 ∩ H2 6= ∅. Let H = H1 ∩ H2, ϑ −1(δ1) = ε1, ϑ−1(δ2) = ε2, and η = ϕ(ε1, ε2). We prove that T(H|Lη) separates the sets UTδ1(H1) and U T δ2 (H2), and s-dm IK,IB IE,ν (T(H|Lη )) ≤ κ − 1 < κ. Since IPκ−1 is a saturated class of spaces, the subspace T(H|Lη ) of T(M|Lη , R|Lη) belongs to IPκ−1. Hence, s-dm IK,IB IE,ν (T(H|Lη )) ≤ κ − 1 < κ. We prove that the subset T(H|Lη ) of T(L) separates U T δ1 (H1) and U T δ2 (H2). Suppose that X ∈ H. Since the subsets V Xε1 and V X ε2 of X are disjoint, by condition (1) we have (k) V Xε1 ⊆ W X η , V X ε2 ⊆ OXη , (l) W Xη ∩ O X η = ∅, and (m) X \ LXη = W X η ∪ O X η . The above relations imply that (n) UTδ1(H) ⊆ T(H|Wη ) = T|Wη ∩ T(H), U T δ2 (H) ⊆ T(H|Oη ) = T|Oη ∩ T(H), (o) T(H|Wη ) ∩ T(H|Oη ) = ∅, and (p) T(H) \ T(H|Lη) = T(H|Wη ) ∪ T(H|Oη ). Since the restriction Wη of S is open and the family R is (M, Co(Wη))- admissible, by Lemma 1.4.7 of [2], the subset T|Wη of T is open. Similarly, Universal elements for some classes of spaces 209 the subset T|Oη of T is open. Also, since the subset T(H) of T is open and T(H) ⊆ T(L), the sets T(H|Wη) and T(H|Oη ) are open in T(L). Setting W = T(H1 \ H) ∪ T(H|Wn) and O = T(L \ H1) ∪ T(H|On) we have (q) UTδ1(H1) ⊆ W , U T δ2 (H2) ⊆ O, (r) W ∩ O = ∅, and (s) T(L) \ T(H|Lη ) = W ∪ O. Therefore, the subset T(H|Lη) of T(L) separates the sets U T δ1 (H1) and U T δ1 (H2). Thus, the class IP(s-Dm IK,IB IE,ν ≤ κ) is saturated. Now, we prove that the class IP(s-dm IK,IB IE,ν ≤ κ) is saturated. Let S be a indexed collection of elements of IP(s-dm IK,IB IE,ν ≤ κ). For every X ∈ S there exists an indexed set {QXi : i ∈ ν} of subsets of X such that (3) X = ∪{QXi : i ∈ ν}. (4) For every i ∈ ν, the subset QXi of X is closed and (Q X i , X) ∈ IK. (5) For every i ∈ ν, s-Dm IK,IB IE,ν (Q X i ) ≤ κ. We set Qi = {Q X i : X ∈ S}, i ∈ ν. By the preceding, the class IP ≡ IP(s-Dm IK,IB IE,ν ≤ κ) is saturated. By property (5), Qi is an indexed collec- tion of elements of the class IP. Therefore, there exists an initial co-mark M+ Qi of Qi corresponding to the class IP. Denote by Mi a co-mark of S such that its trace on Qi is a co-extension of the co-mark M + Qi . By property (4), the restriction Qi of S is a IK-restriction. Since IK is a saturated class of subsets, for every i ∈ ν there exists an initial co-mark M+IK,i of S corresponding to the IK-restriction Qi. By Lemma 2.1.2 of [2], there exists a co-mark M+ of S, which a co-extension of the co-marks Mi and M + IK,i for every i ∈ ν. We show that M + is an initial co-mark of S corresponding to the class IP(s-dm IK,IB IE,ν ≤ κ). Indeed, let M ≡ {{UXδ : δ ∈ τ} : X ∈ S} be an arbitrary co-extension of M+. Then, M is a co-extension of the co-marks Mi and M + IK,i and the co-mark M|Qi is a co-extension of the co-mark M + Qi of Qi, i ∈ ν. Let R+ Qi be an initial family of equivalence relations on Qi corresponding to the co-mark M|Qi and the class IP. Denote by Ri the family of equivalence relations on S such that the trace on Qi of Ri is the family R + Qi . Let also R+IK,i be an initial family of equivalence relations on S corresponding to the co-mark M and the IK-restriction Qi. 210 D. N. Georgiou, S. D. Iliadis and A. C. Megaritis By Lemma 2.1.1 of [2], there exists an admissible family R+ of equivalence relations on S, which is a final refinement of the families Ri and R + IK,i, i ∈ ν. Therefore, R+ is an M-admissible family. We prove that R+ is an initial family of S corresponding to the co-mark M of S and the class IP(s-dm IK,IB IE,ν ≤ κ). For this purpose, we consider an arbitrary admissible family R of equivalence relations on S, which is a final refinement of R+. Then, R is a final refinement of the families Ri and R + IK,i for every i ∈ ν. We need to prove that for every L ∈ C♦(R), T(L) ∈ IP(s-dm IK,IB IE,ν ≤ κ). Let L ∈ C♦(R). It suffices to show that T(L) = ∪{Ti(L) : i ∈ ν} such that (t) the subset Ti(L) of T(L) is closed, (u) (Ti(L), T(L)) ∈ IK, and (v) s-Dm IK,IB IE,ν (Ti(L)) ≤ κ, i ∈ ν. We set Ti(L) = T(L|Qi ), i ∈ ν. It is easy to verify that the subset T(L|Qi ) of T(L) is closed and T(L) = ∪{T(L|Qi) : i ∈ ν}. Since IK is a saturated class of subsets, (T(L|Qi ), T(L)) ∈ IK. Since IP is a saturated class, the subspace T(L|Qi) of T(M|Qi, R|Qi) belongs to IP. Hence, s-Dm IK,IB IE,ν (T(L|Qi)) ≤ κ. Thus, by condition (3) of Definition 4.2, s-dm IK,IB IE,ν (T(L)) ≤ κ proving that the class IP(s-dm IK,IB IE,ν ≤ κ) is saturated. � Corollary 4.7. For every κ ∈ ω in the classes IP(s-dm IK,IB IE,ν ≤ κ) and IP(s-Dm IK,IB IE,ν ≤ κ) there exist universal elements. Corollary 4.8. Let IP be one of the following classes (a) the class of all (completely) regular spaces of weight ≤ τ, (b) the class of all (completely) regular countable-dimensi-onal spaces of weight ≤ τ, (c) the class of all (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (d) the class of all (completely) regular locally finite-dimensional spaces of weight ≤ τ, and (e) the class of all (completely) regular spaces X of weight ≤ τ such that ind(X) ≤ α ∈ τ+. Then, for every κ ∈ ω in the classes IP(s-dm IK,IB IE,ν ≤ κ) ∩ IP and IP(s-Dm IK,IB IE,ν ≤ κ) ∩ IP there exist universal elements. 5. Questions Question 5.1. Does there exists a universal element in the class of all spaces X with dm IK,IB IE,ν (X) ≤ α or in the class of all spaces X with Dm IK,IB IE,ν (X) ≤ α, where α is an ordinal. Universal elements for some classes of spaces 211 Question 5.2. Does there exists a universal element in the class of all spaces X with w-dm IK,IB IE,ν (X) ≤ α or in the class of all spaces X with w-Dm IK,IB IE,ν (X) ≤ α, where α is an ordinal. Question 5.3. Does there exists a universal element in the class of all spaces X with s-dm IK,IB IE,ν (X) ≤ α or in the class of all spaces X with s-Dm IK,IB IE,ν (X) ≤ α, where α is an ordinal. References [1] R. Engelking, Theory of dimensions, finite and infinite, Sigma Series in Pure Mathe- matics, 10. Heldermann Verlag, Lemgo, 1995. viii+401 pp. [2] S. D. Iliadis, Universal spaces and mappings, North-Holland Mathematics Studies, 198. Elsevier Science B.V., Amsterdam, 2005. xvi+559 pp. [3] D. N. Georgiou, S. D. Iliadis and A.C. Megaritis, Dimension-like functions and univer- sality, Topology Appl. 155 (2008), 2196–2201. [4] A. K. O’ Connor, A new approach to dimension, Acta Math. Hung. 55, no. 1-2 (1990), 83–95. [5] R. Pol, There is no universal totally disconnected space, Fund. Math. 79 (1973), 265– 267. (Received April 2011 – Accepted July 2011) D. N. Georgiou (georgiou@math.upatras.gr) Department of Mathematics, University of Patras, 265 04 Patras, Greece. S. D. Iliadis (iliadis@math.upatras.gr) Department of Mathematics, University of Patras, 265 04 Patras, Greece. A. C. Megaritis (megariti@master.math.upatras.gr) Department of Mathematics, University of Patras, 265 04 Patras, Greece. Universal elements for some classes of spaces. By D. N. Georgiou, S. D. Iliadis and A. C. Megaritis