@ Appl. Gen. Topol. 20, no. 1 (2019), 97-108doi:10.4995/agt.2019.10036 c© AGT, UPV, 2019 F-n-resolvable spaces and compactifications Intissar Dahane a, Lobna Dridi b and Sami Lazaar a a Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia. (intissardahane@gmail.com, salazaar72@yahoo.fr) b Department of Mathematics, Tunis Preparatory Engineering Institute. University of Tunis, 1089 Tunis, Tunisia. (lobna dridi 2006@yahoo.fr) Communicated by S. Garćıa-Ferreira Abstract A topological space is said to be resolvable if it is a union of two disjoint dense subsets. More generally it is called n-resolvable if it is a union of n pairwise disjoint dense subsets. In this paper, we characterize topological spaces such that their re- flections (resp., compactifications) are n-resolvable (resp., exactly-n- resolvable, strongly-exactly-n-resolvable), for some particular cases of reflections and compactifications. 2010 MSC: 54B30; 54D10; 46M15. Keywords: categories; functors; resolvable spaces; compactifications. Introduction Let n > 1 be an integer. Generalizing the concept of resolvable spaces introduced by Hewitt in [16], Ceder in [6] defined a topological space X to be n-resolvable space if it has a family of n pairwise disjoints dense subsets. The latter is called exactly n-resolvable if it is n-resolvable but not (n + 1)- resolvable and it is called strongly exactly n-resolvable denoted by SEnR if it is n-resolvable and no empty subset of X is (n + 1)-resolvable. SE1R space is commonly said strongly irresolvable space (abbreviated as SI-space) or hered- itarily irresolvable (see [7] and [13]). Received 25 April 2018 – Accepted 01 February 2019 http://dx.doi.org/10.4995/agt.2019.10036 I. Dahane, L. Dridi and S. Lazaar The theory of categories and functors play an enigmatic role in topology, specially the notion of reflective subcategories. Recently, some authors have been interested by particular functors like T0, S, ρ and FH. In [10], [11] and [8], the authors have characterized topological spaces whose F-reflections are door, submaximal, nodec and resolvable. Some papers, as [5] and [3] were interested in spaces such that their com- pactifications are submaximal, door and nodec. Specially in [2], K. Belaid and M. Al-Hajri have characterized topological spaces such that their one point compactifications (resp., Wallman compactifications) are resolvable. In the first section of this paper, we characterize topological spaces such that their T0-reflections are n-resolvable (resp., exactly n-resolvable, strongly exactly n-resolvable). In the second section, topological spaces, such that their Tychonoff reflec- tions and functionally Hausdorff reflections are n-resolvable (resp., exactly n- resolvable), have been characterized. The third section of this paper is devoted to a characterization of topolog- ical spaces such that their one point compactifications (resp., Wallman com- pactifications) are n-resolvable (resp., exactly n-resolvable, strongly exactly n-resolvable). 1. T0-n-resolvable spaces, T0-exactly-n-resolvable spaces and T0-strongly-exactly-n-resolvable spaces. Let X be a topological space. The T0-reflection of X denoted by T0(X) is defined as follow. Consider the equivalence relation ∼ on X by: x ∼ y if and only if {x} = {y}, for x, y ∈ X. Then the resulting quotient space T0(X) := X/ ∼ is a Kolmogroff space called the T0-reflection of X. Recall that a continuous map q : X −→ Y is said to be a quasihomeomor- phism if U 7−→ q−1(U) (resp., C 7−→ q−1(C) ) defines a bijection O(Y ) −→ O(X) (resp., F(Y ) −→ F(X)), where O(X) (resp., F(X)) is the collection of all open sets (resp., closed sets) of X )[15]. In particular the canonical surjection µX : X −→ T0(X) is an onto quasihome- omorphism and consequently a closed (resp., open) map, (see [4]). In order to give the main result of this section we recall the following results introduced in [10]. Notation 1.1 ([10, Notations 2.2]). Let X be a topological space, a ∈ X and A ⊆ X. We denote by: (1) d0(a) := {x ∈ X : {x} = {a}}. (2) d0(A) = ∪[d0(a); a ∈ A]. Remark 1.2 ([10, Remarks 2.3]). Let X be a topological space and A be a subset of X. The following properties hold. (i) d0(A) = µ −1 X (µX(A)). c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 98 F -n-resolvable spaces and compactifications (ii) d0(d0(A)) = d0(A). (iii) A ⊆ d0(A) ⊆ A and consequently d0(A) = A. (iv) In particular if A is open (resp., closed ), then d0(A) = A. The following definitions are natural. Definition 1.3. A topological space X is called T0-n-resolvable (resp., T0- exactly-n-resolvable, T0-strongly-exactly-n-resolvable) if its T0-reflection is n- resolvable (resp., exactly-n-resolvable, strongly-exactly-n-resolvable). Before giving the characterization of T0-n-resolvable spaces, let us introduce the following definition. Definition 1.4. A family {Ai : i ∈ I} of subsets of a topological space X is called pairwise d0-disjoint if and only if d0(Ai)∩d0(Aj) = ∅, for any i 6= j ∈ I. By Remarks 1.2 (iii), a pairwise d0-disjoint family is a pairwise disjoint family. The following result characterise T0-n-resolvable spaces. Theorem 1.5. Let X be a topological space. Then the following statements are equivalent: (1) X is a T0-n-resolvable space; (2) X have a dense pairwise d0-disjoint family with cardinality n. Proof. (1)=⇒(2) Suppose that X is a T0-n-resolvable space. Then T0(X) has a dense pairwise disjoint family {µX(Ai); 1 ≤ i ≤ n}, where A1,..., An are subsets in X. So applying µ−1 X , one can see easily that {d0(Ai) : 1 ≤ i ≤ n} is a family of pairwise disjoint subsets of X. Now since µ X is an onto quasihomeomorphism then, by [10, Lemma 2.16], we have: ∀1 ≤ i ≤ n X = µ−1 X (T0(X)) = µ −1 X ( µ X (Ai) ) = µ−1 X (µ X (Ai)) = d0(Ai). Therefore {Ai; 1 ≤ i ≤ n} is a dense pairwise d0-disjoint family of X. (2)=⇒(1) Suppose that X has a dense pairwise d0-disjoint family {Ai; 1 ≤ i ≤ n} with cardinality n. Then, for any 1 ≤ i 6= j ≤ n, the condition d0(Ai) ∩ d0(Aj) = ∅ implies immediately that µ X (Ai) ∩ µX (Aj)) = ∅. Now, let 1 ≤ i ≤ n. The density of d0(Ai) in X shows that: T0(X) = µX(X) = µX(d0(Ai)) = µX(µ−1 X (µ X (Ai))) = µX(µ −1 X ( µ X (Ai) ) ) = µX(Ai). Therefore, {µX(Ai) : 1 ≤ i ≤ n} is a dense pairwise disjoint family of subsets of T0(X). � Remark 1.6. Clearly every T0-n-resolvable space is a n-resolvable space. The converse does not hold, indeed: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 99 I. Dahane, L. Dridi and S. Lazaar Let X be a subset of cardinality n (n > 1) equipped with the indiscreet topology. Clearly the family {{x}; x ∈ X} is composed by disjoint dense subsets of X and thus X is n-resolvable. But T0(X) is a one point which is not 2-resolvable. Remark that in this case d0({x}) = X, for any x ∈ X and consequently, d0(A) = X for any subset A of X, therefore there is no d0-disjoint family of X with cardinality greater or equal to 2. The following result is an immediate consequence of the previous theorem. Corollary 1.7. Let X be a topological space. X is a T0-exactly-n-resolvable space if and only if max{| F | F is a dense d0 − disjoint family of X} = n. Before giving a characterization of a T0-strongly-exactly-n-resolvable space we need the following lemma. Lemma 1.8. Let X be a topological space and S a subset of X. Then µX(S) ≃ µS(S). Proof. S is a subset of X then, the following diagram is commutative. S � i // X µX �� T0(S) �� µS T0(i) // T0(X) - T0(i) : T0(S) −→ T0(i)(T0(S)) is bijective. In fact it is enough to show that T0(i) is one-to-one. Let x, y two elements of S such that T0(i)(µS(x)) = T0(i)(µS(y)). Then, µX(i(x)) = µX(i(y)) and thus µX(x) = µX(y). Hence, we get {x} S = {x}∩S = {y} ∩ S = {y} S , as desired. - T0(i) is an open map. Indeed, let Ũ be an open set of T0(S). Then, there exists V an open set of X such that µ−1 S (Ũ) = V ∩ S. Thus T0(i)(Ũ) = T0(i)(µS(V ∩ S)) = µX(i(V ∩ S)) = µX(V ∩ S) So, let us show that µX(V ∩ S) = µX(V ) ∩ T0(i)(T0(S)). Indeed: µX(V ∩ S) ⊆ µX(V ) ∩ µX(S) = µX(V ) ∩ µX(i(S)) = µX(V ) ∩ T0(i)(µS(S)) = µX(V ) ∩ T0(i)(T0(S)) Which gives the first inclusion. Conversely, let x ∈ µX(V ) ∩ T0(i)(T0(S)). Then there exist y ∈ V and t ∈ S such that µX(y) = x = T0(i)(µS(t)) = µX(i(t)) = µX(t). Thus, {y} = {t}. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 100 F -n-resolvable spaces and compactifications Since y ∈ V , then V ∩ {t} 6= ∅. So, x = µX(t) ∈ µX(V ∩ S) which proves that µX(V )∩T0(i)(T0(S)) ⊆ µX(V ∩S) which gives the second inclusion as desired. - µX(S) ≃ µS(S). According to the above, we conclude that T0(i) is an homeomorphism from T0(S) to T0(i)(T0(S)). Then, µX(S) = µX(i(S)) = T0(i)(µS(S)) = T0(i)(T0(S)) ≃ T0(S) = µS(S). � Theorem 1.9. Let X be a topological space. Then the following statements are equivalent: (1) X is a T0-strongly-exactly-n-resolvable space. (2) X is T0-n-resolvable and for any subset S of X, S is not T0-(n + 1)- resolvable. Proof. (1)=⇒(2) Let S be a subset of X. Since X is T0-strongly-exactly-n-resolvable, µX(S) is not (n + 1)-resolvable. Then, by Lemma 1.8, µS(S) = T0(S) is not (n + 1)- resolvable. Therefore, X is a T0-n-resolvable space in which every subset S of X, is not T0-(n + 1)-resolvable. (2)=⇒(1) Let µX(S) be a subset of T0(X), where S be a subset of X. By hypoth- esis, S is not T0 − (n + 1)-resolvable that is T0(S) = µS(S) is not (n + 1)- resolvable. Using Lemma 1.8, µX(S) is not (n + 1)-resolvable. So that every subset µX(S) of T0(X) is not (n + 1)-resolvable and thus T0(X) is strongly- exactly-n-resolvable. � 2. ρ-n-resolvable spaces and F H-n-resolvable spaces Recall that a T1 topological space X is called Tychonoff if for any closed subset F of X and for any x ∈ X not in F there exists a real continuous map f from X to R ( we write f ∈ C(X) ) such that f(x) = 0 and f(F) = {1}. We say that F and x are completely separated. In particular two distinct points in a given Tychonoff space X are said to be completely separated if x and {y} are completely separated. A T1 topological space in which every two distinct points are completely separated, is called functionally Hausdorff space. Give a topological space X. We define the equivalence relation ∼ on X by x ∼ y if and only if f(x) = f(y) for all f ∈ C(X). On the one hand, the set of equivalence classes X/ ∼ equipped with the quotient topology, is a functionally Hausdorff space called the FH-reflection of X. On the other hand, consider ρX the canonical surjection map from X to X/ ∼. Then for any continuous map fα from X to R, there exists a unique map ρ(fα) from X/ ∼ to R satisfying ρ(fα)(ρX(x)) = f(x), for any x ∈ X. So, X/ ∼ equipped with the the topology whose closed sets are of the form ∩[ρ(fα) −1(Fα) : α ∈ I], where fα : X −→ R (resp., Fα) is a continuous map (resp., a closed subset of R), is a a Tychonoff space (see for instance [22]) called the ρ-reflection of X. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 101 I. Dahane, L. Dridi and S. Lazaar We need to introduce and recall some definitions, notations and results. Notation 2.1 ([10, Notation 3.1]). Let X be a topological space, a ∈ X and A a subset of X. We denote by: (1) dρ(a) := ∩[f −1(f({a})) : f ∈ C(X)]. (2) dρ(A) := ∪[dρ(a) : a ∈ A]. Definition 2.2. Let X be a topological space. X is called: (1) ρ-n-resolvable (resp., FH-n-resolvable) space if its ρ-reflection (resp., FH-reflection) is a n-resolvable space. (2) ρ-exactly-n-resolvable (resp., FH-exactly-n-resolvable) space if its ρ- reflection (resp., FH-reflection) is an exactly-n-resolvable space. (3) ρ-strongly-exactly-n-resolvable (resp., FH-strongly-exactly-n-resolvable) space if its ρ-reflection (resp., FH-reflection ) is a strongly-exactly-n- resolvable space. Recall that for a given topological space X and A ⊆ X, A is called a zero- set if there exists f ∈ C(X) such that A = f−1({0}). The complement of a zero-set is called a cozero-set. A space is Tychonoff if and only if the family of zero-sets of the space is a base for the closed sets (equivalently, the family of cozero-sets of the space is a base for the open sets)(see [22, Proposition 1.7]). In [10] it is showen that a closed (resp., open) subset of ρ(X) is of the form ∩[ρ(f)−1({0}) : f ∈ H] (resp., ∪[ρ(f)−1(R⋆) : f ∈ H]) , where H is a collection of continuous maps from X to R. Definition 2.3 ([10, Definition 3.14]). Let X be a topological space, a subset V of X is called: (i) a functionally open subset of X (for short F-open ) if and only if dρ(V ) is open in X. (ii) a functionally dense subset of X (for short F-dense) if and only if for any F-open subset W of X, dρ(V ) meets dρ(W). (iii) ρ-dense, if g(V ) 6= {0} for every nonzero continuous map g from X to R. Definition 2.4. Let X be a topological space and {Ai : i ∈ I} be a family of subsets of X. We say that this family is pairwise dρ-disjoint if and only if dρ(Ai) ∩ dρ(Aj) = ∅, for any i 6= j ∈ I. Theorem 2.5. Let X be a topological space. Then the following statements are equivalent: (i) X is FH-n-resolvable. (ii) X have a F-dense pairwise dρ-disjoint family with cardinality n. Proof. (i) =⇒ (ii) Suppose that X is an FH-n-resolvable space. Then, there exists a family {ρX(A1), ..., ρX(An)} of dense pairwise disjoint subsets of FH(X). c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 102 F -n-resolvable spaces and compactifications Now, applying ρ−1 X , we see easily that the family {A1, ......, An} is pairwise dρ-disjoint. Finally, the equality ρX(Ai) = FH(X) means that Ai is a F-dense subset of X. Therefore, {A1, ......, An} is pairwise dρ-disjoint family of X with cardinality n. (ii) =⇒ (i) Conversely, let {Ai : 1 ≤ i ≤ n} be a family of F-dense pairwise dρ-disjoint subsets of X. Then on the one hand, for every 1 ≤ i ≤ n, ρX(Ai) is a dense subset of FH(X) and on the other hand, ∀ 1 ≤ i 6= j ≤ n, we have dρ(Ai) ∩ dρ(Aj)) = ρ −1 X (ρX(A1)) ∩ ρ −1 X (ρX(Aj))) = ρ−1 X (ρX(Ai) ∩ ρX(Aj))) = ∅ Therefore, {ρX(A1), ..., ρX(An)} is a family of dense pairwise disjoint subsets of FH(X). � By the same way as in Theorem 2.5, the following result is immediate. Theorem 2.6. Let X be a topological space. Then the following statements are equivalent: (i) X is ρ-n-resolvable. (ii) X have a ρ-dense and pairwise dρ-disjoint family of cardinality n. Remark 2.7. Since every F-dense subset is a ρ-dense subset ( see [10, Remarks 3.15] ), then by Theorem 2.6, every FH-n-resolvable space is ρ-n-resolvable. The following results are immediate. Corollary 2.8. Let X be a topological space. X is a FH-exactly-n-resolvable space if and only if max{| F | F is F-dense and dρ−disjoint family of X} = n. Corollary 2.9. Let X be a topological space. X is a ρ-exactly-n-resolvable space if and only if max{| F | F is ρ-dense and dρ−disjoint family of X} = n. Remark 2.10. Regarding Lemma 1.8, this result does not subsist in the case of the functors FH and ρ as showing by the following example. Consider the Alexandroff space X = Z ∪ {∞} such that {n} = {n}, for every n ∈ Z and {∞} = X. It is clear that every real continuous map from X is constant and thus FH(X) = ρ(X) is a one point space. Now, consider S = Z, then FH(S) = ρ(S) = S, but ρX(S) is a one point. One can illustrates this situation by the following picture. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 103 I. Dahane, L. Dridi and S. Lazaar .. . -4 -3 -2 -1 0 1 2 3 4 . . . ∞ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚✚ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩❩ Question 2.11. The Theorem 1.9 is an immediate consequence of Lemma 1.8 which is not valuable in the case of the functors FH and ρ as showing by Remark 2.10. Hence the following question is immediate. Are FH-strongly- exactly-n-resolvable (resp., ρ-strongly-exactly-n-resolvable) spaces equivalent to FH-n-resolvable (resp., ρ-n-resolvable) in which every subset S of X, is not FH-(n + 1)-resolvable (resp., ρ-(n + 1)-resolvable)? 3. n-resolvable spaces and compactifications Definition 3.1. A compactification of a topological space X is a pair (K(X), e) where K(X) is a compact space and e an embedding of X as a dense subset of K(X). Remark 3.2. In many cases, e will be an inclusion map, so that X ⊆ K(X). In other cases , we can agree to write X when mean e(X), so that we can again write X ⊆ K(X). Whenever one of this situations occurs we say simply that K(X) is a compactification of X, and think of K(X) as containing X as a dense subspace. Lemma 3.3 ([2, Lemma 2.1]). Let X be a topological space and K(X) be a compactification of X and A be a subset of K(X). If X is an open set of K(X) Then the following statements are equivalent: (1) A is a dense subset of K(X). (2) A ∩ X is a dense subset of X. Using Lemma 3.3, the following proposition is immediate. Proposition 3.4. Let X be a topological space and K(X) be a compactification of X. If X is an open set of K(X) Then the following statements are equivalent: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 104 F -n-resolvable spaces and compactifications (1) X is n-resolvable. (2) K(X) is n-resolvable. Recall that for a topological space X, the set X̃ = X∪{∞} with the topology whose members are the open sets of X and all subsets U of X̃ such that X̃ \ U is a closed compact subset of X, is called the Alexandroff extension of X ( or the one-point compactification of X ). Now, regarding Proposition 3.4, we get immediately the following result. Corollary 3.5. Let X be a non compact topological space Then the following statements are equivalent: (1) The one point compactification X̃ of X is n-resolvable. (2) X is n-resolvable. We turn our attention to spaces such that their Wallman compactifications are n-resolvable spaces. First, let us recall the construction of Wallman compactification of T1-space (a concept introduced, in 1938, by Wallman [23]). Let P be a class of subsets of a topological space X wich is closed under fnite intersections and finite unions. A P-filter on X is a collection F of nonempty elements of P with the prop- erties: (a) P1, P2 ∈ F implies P1 ∩ P2 ∈ F. (b) P1 ∈ F P1 ⊆ P2 implies P2 ∈ F. A P-ultrafilter is a maximal P-filter. When P is the class of closed sets of X, then the P-filters are called closed filters. The points of the Wallman compactification wX of a space X are the closed ultrafilters on X. For each closed set D ⊆ X, define D∗ to be the set D∗ = {A ∈ wX : D ∈ A}, if D 6= ∅ and ∅∗ = ∅. Thus C = {D∗ : D is a closed set of X} is a base for the closed sets of a topology on wX. Let U be an open subset of X. We define U∗ = {A ∈ wX : F ⊆ U for some F in A}. It is easily seen that the class {U∗ : U is an open set of X} is a base for the open sets of the topology of wX. The following properties of wX are frequently useful: Proposition 3.6. For x ∈ X, let w X (x) = {A | A is a closed set of X and x ∈ A}. Then wX is an embedding of X into wX. Thus, if x ∈ X, then wX (x) will be identified to x. Proposition 3.7. If U ⊂ X is open, then wX \ U∗ = (X \ U)∗. Proposition 3.8. If D ⊂ X is closed, then wX \ D∗ = (X \ D)∗. Proposition 3.9. If U1 and U2 are open in X, then (U1 ∩ U2) ∗ = U∗1 ∩ U ∗ 2 and (U1 ∪ U2) ∗ = U∗1 ∪ U ∗ 2 . In [19], Kovar has characterized space with finite Wallman compactification remainder as following: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 105 I. Dahane, L. Dridi and S. Lazaar Proposition 3.10. Let X be a T1-space, wX the Wallman compactification of X and k a finite number. Then the following statements are equivalent: (1) Card(wX − X) = k. (2) There exists a collection of k pairwise disjoint non compact closed sets of X and every family of non compact pairwise disjoint closed sets of X contain at most k elements. The following proposition follows immediately from Proposition 3.10. Proposition 3.11. Let X be a T1-space and k ∈ N such that every family of non compact pairwise disjoint closed sets of X contain at most k elements. Then X is n-resolvable if and only if wX is n-resolvable. Corollary 3.12 ([2, corollary 3.5]). Let X be a T1-space, wX be the Wallman compactification of X and U be an open set of X. Then the following statements are equivalent: (1) U U∗. (2) There exists a non compact closed set F of X such that F ⊆ U. Definition 3.13. Let X be a T1-topological space. Then X is said to be w-n-resolvable, if its Wallman compactification is n-resolvable. Before characterizing w-n-resolvable spaces, let us introduce the useful def- inition. Definition 3.14. We said that a finite family of subsets {Di, i ∈ I} of a topological space (X, O(X)) satisfies the property (P) if: for every (i, O) ∈ J = I × {O ∈ O(X) : O ∩ Di = ∅}, there exists a non compact closed subset FO,i ⊂ O with {FO,i : (i, O) ∈ J} is a family of pairwise disjoint subsets of X. Now, let us give one of the main result of this section. Theorem 3.15. Let X be a T1- topological space, Then the following state- ments are equivalent: (1) X is w-n-resolvable. (2) X is a partition of a family of n subsets satisfying (P). Proof. Let X be a w-n-resolvable space. Then there exist n pairwise disjoint dense subsets A1,A2,..., An of wX such that wX = A1 ∪ A2 ∪ ... ∪ An. We de- note Di = Ai∩X. It is clear that the family {Di; 1 ≤ i ≤ n} is a partition of X. Let O be a nonempty open subset of X such that O ∩ Di = ∅. The density of Ai in wX gives an element Fi ∈ O ∗ ∩ Ai. By Corollary 3.12, there exists a non compact closed subset G(i,O) ⊂ O such that G(i,O) ∈ Fi. Now, if i′ is distinct from i and O′ is a given nonempty open subset of X such that O′ ∩Di′ = ∅, by the same way, there exists an element Fi′ ∈ O ′∗ ∩Ai′ and consequently there exists a non compact closed subset G(i′,O′) ⊂ O ′ such that G(i′,O′) ∈ Fi′. Since Ai ∩ Ai′ = ∅, then Fi 6= Fi′. Thus, there exist a c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 106 F -n-resolvable spaces and compactifications closed subsets Fi ∈ Fi and Fi′ ∈ Fi′ such that Fi ∩ Fi′ = ∅. Let F(i,O) = G(i,O) ∩ Fi and F(i′,O′) = G(i′,O′) ∩ F ′ i . It is clear that F(i,O) ∈ Fi ∈ wX \ X and F(i′,O′) ∈ Fi′ ∈ wX \ X. Hence, F(i,O) and F(i′,O′) are non compact closed subsets ( see [2, Lemma 3.4]), which are disjoint. Conversely, let {Di; 1 ≤ i ≤ n} be a partition of X by n subsets satisfying (P). For every 1 ≤ i ≤ n, set Ai = Di ∪ {F ∈ wX − X : F(i,O) ∈ F}, where O is an open subset of X such that O ∩Di = ∅ (it is clearly seen that if Ai = Di, then Di is dense in wX). Clearly, by construction, Ai is a dense subset of wX for every 1 ≤ i ≤ n. To finish, let us show that the family {Ai : 1 ≤ i ≤ n} are pairwise disjoint. So, suppose the existence of 1 ≤ i 6= j ≤ n such that Ai ∩ Aj 6= ∅. Since Di ∩ Dj = ∅, then Ai ∩ Aj ∩ (wX − X) 6= ∅. By construction of Ai and Aj, there exist an ultrafilter Fi ∈ Ai and Fj ∈ Aj such that Fi = Fj. Furthermore, there exist open subsets O, O′ and non compact closed subsets F(i,O) ∈ Fi and F(j,O′) ∈ Fj such that O ∩ Di = ∅, O ′ ∩ Dj = ∅, F(i,O) ⊂ O and F(j,O′) ⊂ O ′. Hence, by the property (P), F(i,O) ∩ F(j,O′) = ∅ and consequently Fi 6= Fj, which leads to a contradiction. � As an immediate consequence of Theorem 3.15, for the particular case when n = 2, we have the following corollary. Corollary 3.16 ([2, Theorem 3.6]). Let X be a T1- topological space, Then the following statements are equivalent: (1) X is w-resolvable. (2) X is a partition of two subsets {D1, D2} and for each nonempty open subset O ⊆ Di (i ∈ {1, 2}), there exists a non compact closed subset F such that F ⊆ O. To close this section the following result is immediate. Corollary 3.17. Let X be a T1-topological space. X is w-exactly-n-resolvable if and only if max{| F | ; F is a partition of X, of n dense subsets satisfying (P)} = n. Acknowledgements. 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