@ Appl. Gen. Topol. 23, no. 2 (2022), 255-268 doi:10.4995/agt.2022.15745 © AGT, UPV, 2022 Partial actions of groups on hyperspaces Luis Mart́ınez , Héctor Pinedo and Edwar Ramirez Escuela de Matemáticas, Universidad Industrial de Santander, Colombia (luchomartinez9816@hotmail.com, hpinedot@uis.edu.co, edwar5119@gmail.com) Communicated by F. Lin Abstract Let X be a compact Hausdorff space. In this work we translate par- tial actions of X to partial actions on some hyperspaces determined by X, this gives an endofunctor 2− in the category of partial actions on compact Hausdorff spaces which generates a monad in this cate- gory. Moreover, structural relations between partial actions θ on X and partial actions determined by 2θ as well as their corresponding globalizations are established. 2020 MSC: 54H15; 54B20; 54F16. Keywords: partial action; globalization; hyperspace; monad. 1. Introduction Given an action µ : G×Y → Y of a group G on a set Y and an invariant subset X of Y (i.e., µ(g,x) ∈ X, for all x ∈ X, and g ∈ G), the restriction of µ to G×X determines an action of G on X. However, if X is not invariant, we obtain a partial action on X. This is a collection of partially defined maps θg (g ∈ G) on X satisfying θ1 = idX and θgh is an extension of the composition θg◦θh, for all g,h ∈ G. The notion of partial action of a group was introduced by R. Exel in [6, 7] motivated by problems arising from C∗-algebras. Since then partial group actions have appeared in many different contexts, such as the theory of operator algebras, algebra, the theory of R-trees, tilings and model theory (see for instance [10]). In topology, partial actions on topological spaces consist of a family of homeomorphism between open subsets of the space, and have been considered in the context of Polish spaces (see [13, 14]), 2-cell complexes (see Received 07 June 2021 – Accepted 14 December 2021 http://dx.doi.org/10.4995/agt.2022.15745 https://orcid.org/0000-0002-3957-3119 https://orcid.org/0000-0003-4432-419X https://orcid.org/0000-0003-4919-9439 J. L. Mart́ınez , H. Pinedo and E. Ramirez [16]), topological semigroups [3] and recently in [11] where introduced in the realm of profinite spaces. It seems that when a partial action on some structure is given, one of the most relevant problems is the question of the existence and uniqueness of a globalization, that is, if a partial action can be realized as restrictions of a corresponding collection of total maps on some superspace. In the topological context, this problem was studied by Abadie [1] and independently by Kellen- donk and Lawson [10]. It was proved that for any continuous partial action θ of a topological group G on a topological space X, there is a topological space Y and a continuous action µ of G on Y such that X is a subspace of Y and θ is the restriction of µ to X. Such a space Y is called a globalization of X. They also show that there is a minimal globalization XG called the enveloping space of X (see subsection 2.2 for details). Recent topological advances on partial actions on (locally) compact spaces include the groupoid approach to the en- veloping spaces associated to partial actions of countable discrete groups [9]. Also several classes of C∗-algebras can be described as partial crossed products that correspond to partial actions of discrete groups on profinite spaces; for instance the Carlsen-Matsumoto C∗ -algebra OX of an arbitrary subshift X (see [5]). The interested reader may consult [4] and [8] for a detailed account in developments around partial actions. On the other hand, the study of hyperspaces has developed for more than one hundred years, topological properties in hyperspaces: dimension, shape, con- tractibility, admissibility, unicoherence, etc., have been topics where researchers have dedicated a lot of attention recently. Furthermore, there are many papers in different areas of mathematics focused on the study of set-valued functions where hyperspaces are the natural environment to work. For instance, in [2] the authors study when a hyperspace can be embedded in a cell or when a cell can be embedded in a hyperspace. Topics concerning the n-od problem, Whitney properties and Whitney-reversible properties have been widely considered, for a detailed account on hyperspaces the interested reader may consult [12] and the reference therein. This work is structured as follows. After the introduction in Section 2 we present the preliminary notions on topological partial actions and their en- veloping actions, at the end of this section we fix a compact Hausdorff space X and state our conventions, notations and results on the hyperspaces H1, H2 and H3 consisting of compact, compact and connected, and finite subsets of X, respectively. In Section 3 we translate partial actions θ of X to partial actions 2θ on H ∈ {H1,H2,H3} and present in Theorem 3.2 and Proposition 3.5 some structural properties preserved by this correspondence. Separation properties relating enveloping actions of θ and 2θ are considered in Corollary 3.12 and Theorem 3.14. Finally, Section 4 has a categorical flavor, where it is considered the category G y CH whose objects are topological partial actions on compact Hausdorff spaces and show in Theorem 4.3 that the functor 2− generates a monad in this category. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 256 Partial actions of groups on hyperspaces 2. The notions We present the necessary background on partial actions and hyperspaces that we use throughout the work. 2.1. Preliminaries on partial actions and their enveloping actions. We start with the following. Definition 2.1 ([10, p. 87-88]). Let G be a group with identity element 1 and X be a set. A partially defined function θ : G × X 99K X, (g,x) 7→ g · x is called a (set theoretic) partial action of G on X if for each g,h ∈ G and x ∈ X the following assertions hold: (PA1) If ∃g ·x, then ∃g−1 · (g ·x) and g−1 · (g ·x) = x, (PA2) If ∃g · (h ·x), then ∃(gh) ·x and g · (h ·x) = (gh) ·x, (PA3) ∃1 ·x and 1 ·x = x, where ∃g ·x means that g ·x is defined. We say that θ acts (globally) on X or that θ is global if ∃g ·x, for all (g,x) ∈ G×X. Given a partial action θ of G on X, g ∈ G and x ∈ X. We set: • G∗X = {(g,x) ∈ G×X | ∃g ·x} the domain of θ. • Xg = {x ∈ X | ∃g−1 ·x}. Then θ induces a family of bijections {θg : Xg−1 3 x 7→ g · x ∈ Xg}g∈G. We also denote this family by θ. The following result characterizes partial actions in terms of a family of bijections. Proposition 2.2 ([15, Lemma 1.2]). A partial action θ of G on X is a family θ = {θg : Xg−1 → Xg}g∈G, where Xg ⊆ X, θg : Xg−1 → Xg is bijective, for all g ∈ G, and such that: (i) X1 = X and θ1 = idX; (ii) θg(Xg−1 ∩Xh) = Xg ∩Xgh; (iii) θgθh : Xh−1 ∩Xh−1g−1 → Xg∩Xgh, and θgθh = θgh in Xh−1 ∩Xh−1g−1 ; for all g,h ∈ G. In view of Proposition 2.2 a partial action on X are frequently denoted as a family of maps (θg,Xg)g∈G, between subsets of X satisfying conditions (i)-(iii) above. For the reader’s convenience we recall a characterization of partial action. Proposition 2.3 ([8, Proposition 2.5]). Let G be a group and X a set. Then a family θ = {θg : Xg−1 → Xg}g∈G, of bijections between subsets of X is a partial action of G on X if and only if, in addition to (i) of Proposition 2.2, for all g,h ∈ G one has that: (ii’) θg(Xg−1 ∩Xh) ⊆ Xgh, (iii’) θg(θh(x)) = θgh(x), for all x ∈ Xh−1 ∩X(gh)−1. From now on in this work G will denote a topological group and X a topo- logical space. We endow G×X with the product topology and G∗X with the topology of subspace. Moreover θ : G ∗ X → X will denote a partial action. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 257 J. L. Mart́ınez , H. Pinedo and E. Ramirez We say that θ is a topological partial action if Xg is open and θg is a homeo- morphism, for all g ∈ G. Moreover, if θ is continuous, θ is called a continuous partial action. 2.2. Restriction of global actions and globalization. Let µ: G×Y → Y be a continuous action of G on a topological space Y and X ⊆ Y be an open set. Then we can obtain by restriction a topological partial action on X by setting: (2.1) Xg = X ∩µg(X), θg = µg � Xg−1 and θ: G∗X 3 (g,x) 7→ θg(x) ∈ X. Then θ is a topological partial action of G on X, we say that θ is the restriction of µ to X. As mentioned in the introduction, a natural problem in the study of partial actions is whether they can be restrictions of global actions. In the topological sense, this turns out to be affirmative and a proof was given in [1, Theorem 1.1] and independently in [10, Section 3.1]. Their construction is as follows. Let θ be a topological partial action of G on X and consider the following equivalence relation on G×X: (2.2) (g,x)R(h,y) ⇐⇒ x ∈ Xg−1h and θh−1g(x) = y. Denote by [g,x] the equivalence class of (g,x). The enveloping space or the globalization of X is the set XG = (G × X)/R endowed with the quotient topology. We have by [1, Theorem 1.1] that the action (2.3) µ: G×XG 3 (g, [h,x]) → [gh,x] ∈ XG, is continuous and is the so called the enveloping action of θ. Further by (ii) in [10, Proposition 3.9] the map q : G×X 3 (g,x) 7→ [g,x] ∈ XG, is open. On the other hand the map (2.4) ι: X 3 x 7→ [1,x] ∈ XG satisfies G · ι(X) = XG. Moreover, it follows by [10, Proposition 3.12] that ι a homeomorphism onto ι(X) if and only if θ is continuous, and by [10, Proposition 3.11] ι(X) is open in XG, provided that G∗X is open. We finish this section with a result that will be useful in the sequel. Lemma 2.4. Let µ : G × Y → Y be a continuous global action of G on a topological space Y and let U ⊆ Y be such that G ·U = Y . Then the following assertions hold. (i) If G and U are separable, then Y is separable. (ii) If U is clopen and regular, then Y is regular. Proof. (i) Let {un : n ∈ N} ⊆ U and {gm : m ∈ N} ⊆ G be dense subsets of U and G, respectively. Then for an open nonempty set V ⊆ Y we have that W := µ−1(V ) ∩ (G × U) is open in G × U. Then there are n,m ∈ N © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 258 Partial actions of groups on hyperspaces such that (gm,un) ∈ W and consequently, gm · un ∈ V which implies that {gm ·un ∈ Y : m,n ∈ N} is dense in Y . (ii) Take y ∈ Y and Z ⊆ Y an open set such that y ∈ Z. The fact that G ·U = Y implies that there are g ∈ G, u ∈ U such that y = g ·u. Since µ is continuous there is an open set B ⊆ Y for which u ∈ B and g ·B ⊆ Z. Then V = U ∩B is open in U and g ·V ⊆ Z. Since U is regular, there is an open set W of U such that u ∈ W ⊆ ClU (W) ⊆ V but U is closed then y = g ·u ∈ g ·W ⊆ g ·ClU (W) = g ·W ⊆ g ·V ⊆ Z, and Y is regular. � 2.3. Conventions on hyperspaces. From now on in this work X will denote a compact Hausdorff space. The hyperspace H1 := 2X is the set consisting of non-empty compact subsets of X. For U1,U2, · · · ,Un non-empty open sets of X, let 〈U1, ...,Un〉H1 = { A ∈H1 : A ⊆ n⋃ i=1 Ui, and A∩Ui 6= ∅, 1 ≤ i ≤ n } , moreover we set 〈∅〉 := ∅. The Vietoris topology on H1 is generated by collec- tions of the form 〈U1, ...,Un〉H1. We shall also work with the subspaces H2 := {C ∈H1 | C is connected} and H3 := {F ∈H1 | F is finite} that is 〈U1, · · · ,Un〉Hi := Hi∩〈U1, · · · ,Un〉H1, for U1,U2, · · · ,Un open subsets of X and i = 2, 3. Finally, when taking about a hyperspace H we make reference to any of the spaces H1,H2 as well as H3. We summarize some well-known properties of the space H. For more details on hyperspaces, the interested reader may consult [12]. Lemma 2.5. Let X be a compact Hausdorff space. Then the following asser- tions hold. (i) The map X 3 x 7→ {x}∈H is an embedding of X into H. (ii) H is a compact Hausdorff space and the map u : 22 X → 2X, A 7→ ∪A is continuous. 3. From partial actions on X to partial actions on H In what follows we shall use a continuous partial action on X to construct a continuous partial action on H. The following is straightforward. Lemma 3.1. Let U and V be open subsets of X and f : U → V a homeomor- phism, then the map 2f : 〈U〉H 3 A 7→ f(A) ∈ 〈V 〉H is a homeomorphism. Theorem 3.2. Let θ := (θg,Xg)g∈G be a topological partial action of G on X. For g ∈ G, we set 2θg : 〈Xg−1〉H 3 A 7→ θg(A) ∈ 〈Xg〉H. Then 2θ = (2θg,〈Xg〉H)g∈G is a topological partial action of G on H and the following assertions hold. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 259 J. L. Mart́ınez , H. Pinedo and E. Ramirez (i) G∗H is open provided that G∗X is open. (ii) If θ is continuous, then 2θ is continuous. (iii) If θ is global then 2θ is global. Proof. We shall only deal with the case H = H2. By Lemma 3.1 we have that 2θg is a homeomorphism between open subsets of H2, for any g ∈ G. We shall check conditions (i) and (ii’) - (iii’) in Proposition 2.2 and Proposition 2.3, respectively. To see (i) notice that 2θe is the identity map of 〈X〉H2 = H2. For (ii’) take g,h ∈ G and A ∈ 〈Xg−1〉H2 ∩ 〈Xh〉H2 = 〈Xg−1 ∩ Xh〉H2 , then 2θg (A) = θg(A) ⊆ θg(Xg−1 ∩Xh) ⊆ Xgh, and thus 2θg (〈Xg−1〉H2 ∩〈Xh〉H2 ) ⊆ 〈Xgh〉H2 . For (iii’) take A ∈ 〈Xh−1〉H2 ∩〈X(gh)−1〉H2 = 〈Xh−1 ∩ X(gh)−1〉H2 , then 2θgh(A) = θgh(A) = θg(θh(A)) = 2 θg (2θh(A)), and we conclude that 2θ is a partial action of G on H2. Now we check (i)−(iii). (i) Suppose that G ∗ X is open in G × X. To see that G ∗ H2 is open in G × H2, take (g,A) ∈ G ∗ H2. Since A ⊆ Xg−1 , we have (g,a) ∈ G ∗ X for all a ∈ A. Now the fact that G ∗ X is an open subset of G × X, implies that for any a ∈ A there are open sets Ua ⊆ G and Va ⊆ X for which (g,a) ∈ Ua × Va ⊆ G ∗ X. Since A is compact, there exist a1, · · · ,an ∈ A with A ⊆ n⋃ i=1 Vai, and A ∈ 〈Va1, · · · ,Van〉H2 . Let U := n⋂ i=1 Uai, then (g,A) ∈ U×〈Va1, · · · ,Van〉H2 we claim that U×〈Va1, · · · ,Van〉H2 ⊆ G∗H2. Indeed, take (h,B) ∈ U×〈Va1, · · · ,Van〉H2, we shall check B ∈ 〈Xh−1〉H2 . Take b ∈ B. Since B ⊆ n⋃ i=1 Vai, there is 1 ≤ i ≤ n for which b ∈ Vai and (h,b) ∈ Uai×Vai ⊆ G∗X, then b ∈ Xh−1 . From this we get B ∈ 〈Xh−1〉H2 and thus (h,B) ∈ G ∗H2. This shows that G∗H2 is open in G×H2. (ii) Suppose that θ is continuous. We need to show that 2θ : G∗H2 →H2, (g,A) 7→ θg(A) is continuous. Let (g,A) ∈ G ∗H2 and take V1, · · · ,Vk open subsets of X such that θg(A) ∈ 〈V1, · · · ,Vk〉H2 . For each a ∈ A there is 1 ≤ ia ≤ k such that θg(a) ∈ Via, and since θ is continuous there are open sets Uia ⊆ G and Wia ⊆ X such that: (g,a) ∈ (Uia ×Wia) ∩ (G∗X) and θ((Uia ×Wia) ∩G∗X) ⊆ Via. The fact that A is compact implies that there are a1, · · · ,am ∈ A such that A ⊆ m⋃ j=1 Wiaj . On the other hand, since θg(A) ∩ Vj 6= ∅, for any 1 ≤ j ≤ k, there are r1,r2, · · · ,rk ∈ A for which θg(rj) ∈ Vj, for all j = 1, 2, · · · ,k. Set irj := j, j = 1, · · · ,k. Without loss of generality we may suppose {r1,r2, · · · ,rk} ⊆ {a1, · · · ,am} and ri = ai, for each i = 1, · · · ,k. Let U := ⋂m j=1 Uiaj . Then (g,A) ∈ Z := (U ×〈Wia1 , · · · ,Wiam〉H2 ) ∩ (G ∗H2). To finish the proof it is enough to show that 2θ(Z) ⊆ 〈V1, · · · ,Vk〉H2 . For this take (h,B) ∈ Z and © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 260 Partial actions of groups on hyperspaces b ∈ B. Since B ⊆ m⋃ j=1 Wiaj , there exists 1 ≤ j ≤ m such that b ∈ Wiaj and thus (h,b) ∈ Uiaj ×Wiaj . But B ⊆ Xh−1 , then (h,b) ∈ (Uiaj ×Wiaj ) ∩ (G∗X) and θh(b) ∈ Viaj which implies θh(B) ⊆ k⋃ i=1 Vi. Finally, for 1 ≤ l ≤ k we see that θh(B) ∩Vl 6= ∅. Indeed, take b ∈ B ∩Wial , where al = rl. Since h ∈ Uial , we have (h,b) ∈ (Uial ×Wial )∩(G∗X) and θh(b) ∈ Vial ∩θh(B) = Virl ∩θh(B) = Vl ∩θh(B) which finishes the proof of the second item. (iii) This is clear. � Remark 3.3. Given a partial action θ of G on X, we shall refer to 2θ as the induced partial action of θ on H. Example 3.4. There is a topological partial action of Z(4) on S1 given by the family {Xn}n∈Z(4) as it is shown below. X1X3 X2 θ0 = IdS1 ; θ1 : X3 → X1 by θ1(eit) = ei(t+π); θ3 = θ−11 , θ2 : X2 → X2 is the identity. We construct the induced partial action of Z(4) on H2(S1), for this we find a homeomorphism h between H2(S1), the connected sets of S1 and D = {z ∈ C : |z| ≤ 1}. O h(A) P A Let P ∈S1 and take an arc center at P of length l, this arc is mapped on h(A) =( 1− l 2π ) P ∈ D. The arc {P} of length zero is mapped onto h({P}) = P ∈ D. In particular, all arcs centered at P are mapped on OP. From this follows that the sets {〈Xn〉H2}n∈Z(4) are © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 261 J. L. Mart́ınez , H. Pinedo and E. Ramirez 〈X1〉H2〈X3〉H2 〈X2〉H2 We construct 2θ. The map 2θ2 is the identity on 〈X2〉H2 . Notice that 2θ1 rotates each arc in X3 π radians to an arc in X1 of the same length. X1X3 X2 A 2θ1 (A) h(A) h(2θ1 (A)) Then h(2θ1 (A)) is obtained by rotat- ing h(A) π radians, from this 2θ1 in D is identified with 2θ1 : 〈X3〉H2 −→ 〈X1〉H2 , reit 7−→ rei(t+π), analogously 2θ3 : 〈X1〉H2 −→ 〈X3〉H2 , reit 7−→ rei(t+π). We finish this section with the next. Proposition 3.5. Let θ be a topological partial action of G on X. If G∗X is closed, then G∗H is closed. Proof. Take (g,A) ∈ (G ×H) \ G ∗H. Then there is a ∈ A such that @g · a and (g,a) /∈ G ∗ X and there are open sets U ⊆ G and V ⊆ X such that (g,a) ∈ U × V ⊆ (G × X) \ G ∗ X. Note that (g,A) ∈ U × 〈V,X〉H to finish the proof we need to show that U ×〈V,X〉H ⊆ (G×H) \G∗H. Take (h,B) ∈ U ×〈V,X〉H and b ∈ B ∩V . Since (h,b) ∈ U ×V , we get @h · b, then @h ·B and (h,B) /∈ G∗H as desired. � 3.1. Separation properties and enveloping spaces. It is shown in [1, Proposition 1.2] that a partial action has a Hausdorff enveloping space if and only if the graph of the action is closed. Below we show that partial actions on compact Hausdorff spaces have Hausdorff enveloping space, if and only if the enveloping space of the induced partial action on H is Hausdorff. From now on, 2R denotes the equivalence relation associated to the envelop- ing action of the partial action 2θ of G on H (see equation (2.2)). That is HG = (G×H)/2R. Lemma 3.6. Let θ be a partial action on X and 2θ be the corresponding partial action of G on H, then the map Θ : XG 3 [g,x] 7→ [g,{x}] ∈HG is an embedding. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 262 Partial actions of groups on hyperspaces Proof. First of all observe that Θ is well defined. Indeed, if (g,x)R(h,y), then {x} ⊆ 〈Xg−1h〉H and 2 θ h−1g ({x}) = {y}, which gives (g,{x})2R(h,{y}). In an analogous way one checks that Θ is injective. Now we prove that Θ is continuous, for this it is enough to check that β : G×X 3 (g,x) 7→ [g,{x}] ∈ HG, is continuous. For this notice that ϕ : G×X 3 (g,x) 7→ (g,{x}) ∈ G×H, is continuous because of Lemma 2.5. Also, β = qH◦ϕ, where qH : G×H→HG is the quotient map, form this β is continuous, and so is Θ. Now we need to show that Θ−1 : Im(Θ) → XG is continuous. Let U ⊆ XG be an open set and [g0,{x0}] ∈ Im(Θ) such that [g0,x0] ∈ U. Then (g0,x0) ∈ q−1(U) and there exists open sets V ⊆ G and W ⊆ X such that (g0,x0) ∈ V × W ⊆ q−1(U). Take Z := qH(V × 〈W〉H) ∩ Im(Θ). Since qH is open, then Z is open in Im(Θ) and [g0,{x0}] ∈ Z. On the other hand, take [r,{s}] ∈ Z we check that Θ−1([r,{s}]) = [r,s] ∈ U. For this take (v,F) ∈ V ×〈W〉H such that [v,F] = qH(v,F) = [r,{s}]. Then F = {w} for some w ∈ W and Θ−1([r,{s}]) = Θ−1([v,{w}]) = [v,w] = q(v,w) ∈ q(V ×W) ⊆ U, this shows that Θ−1 is continuous and Θ is an embedding. � Lemma 3.7. Let θ be a partial action on X and 2θ be the induced partial action of G on H, then 2R is closed in (G×H)2 provided that R is closed in (G×X)2. Proof. Take ((g,A), (h,B)) ∈ (G×H)2 \ 2R, we have two cases to consider: Case 1: A /∈ 〈Xg−1h〉H. Then there exists a ∈ A ∩ (X \ Xg−1h), and ((g,a), (h,b)) ∈ (G × X)2 \ R, for any b ∈ B. Since R is closed there are open sets Ub,Yb ⊆ G and Vb,Zb ⊆ X such that ((g,a), (h,b)) ∈ (Ub ×Vb) × (Yb ×Zb) ⊆ (G×X)2 \R, for any b ∈ B. The fact that B is compact implies that there are b1, · · · ,bn ∈ B for which B ⊆ n⋃ i=1 Zbi. Write U := n⋂ i=1 Ubi, V := n⋂ i=1 Vbi and Y = n⋂ i=1 Ybi. Then A ∈ 〈X,V 〉H and ((g,A), (h,B)) ∈ (U ×〈X,V 〉H) × (Y ×〈Zb1, · · · ,Zbn〉H). Now we show that (U ×〈X,V 〉H) × (Y ×〈Zb1, · · · ,Zbn〉H) ⊆ (G×H) 2 \ 2R. For this take ((r,C), (s,D)) ∈ (U ×〈X,V 〉H) × (Y ×〈Zb1, · · · ,Zbn〉H). For c ∈ C∩V and d ∈ D, there is 1 ≤ j ≤ n such that d ∈ Zbj , then ((r,c), (s,d)) ∈ (Ubj ×Vbj )×(Ybj ×Zbj ) ⊆ (G×X)2 \R which implies c /∈ Xr−1s or c ∈ Xr−1s and θs−1r(c) 6= d. If c /∈ Xr−1s, then C /∈ 〈Xr−1s〉H and we have done. Now suppose c ∈ Xr−1s and θs−1r(c) 6= d. If θs−1r(c) ∈ D, by a similar argument as above we get ((r,c), (s,θs−1r(c)) /∈ R, which leads to a contradiction. Then, θs−1r(C) 6= D and ((r,C), (s,D)) /∈ 2R. Case 2. A ⊆ Xg−1h. Then θh−1g(A) 6= B. Suppose that there exists a ∈ A such that θh−1g(a) /∈ B. Then ((g,a), (h,b)) /∈ R, for any b ∈ B, we argue as in Case 1 to obtain b1, · · · ,bn ∈ B and families {Ubi}ni=1, {Ybi} n i=1 of open subsets © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 263 J. L. Mart́ınez , H. Pinedo and E. Ramirez of G such that g ∈ U := n⋂ i=1 Ubi and h ∈ Y := n⋂ i=1 Ybi. Also there are families {Vbi}ni=1 and {Zbi} n i=1 of open subsets of X such that a ∈ V := ⋂ i=1 Vbi,B ∈ 〈Zb1, · · · ,Zbn〉H and (Ubi×Vbi)×(Ybi×Zbi) ⊆ (G×X)2\R, for any i = 1, · · ·n. As in Case 1 we get ((g,A), (h,B)) ∈ (U ×〈X,V 〉H) × (Y ×〈Zb1, · · · ,Zbn〉H) ⊆ (G×H)2 \ 2R. To finish the proof, suppose that there is b ∈ B such that θh−1g(a) 6= b, for each a ∈ A. If a ∈ A, then ((g,a), (h,b)) /∈ R and there are open sets Ua,Ya ⊆ G and Va,Za ⊆ X such that ((g,a), (h,b)) ∈ (Ua×Va)×(Ya×Za) ⊆ (G×X)2\R. The compactness of A implies that there are a1, · · · ,an ∈ A such that A ⊆ n⋃ i=1 Vai. Write U := n⋂ i=1 Uai, Y ′ := n⋂ i=1 Yai and Z := n⋂ i=1 Zai. Now ((g,A), (h,B)) ∈ (U ×〈Va1, · · · ,Van〉H) × (Y ′ ×〈X,Z〉H) ⊆ (G×H)2 \ 2R. Indeed, let ((r,C), (s,D)) ∈ (U ×〈Va1, · · · ,Van〉H) × (Y ′ ×〈X,Z〉H) and d ∈ D∩Z. For c ∈ C, there is 1 ≤ j ≤ n such that c ∈ Vaj therefore ((r,c), (s,d)) ∈ (Uaj × Vaj ) × (Yaj × Zaj ) ⊆ (G × X)2 \ R. Moreover, ((s,d), (r,c)) /∈ R and d /∈ Xs−1r or d ∈ Xs−1r and θr−1c(d) 6= c. In the case d /∈ Xs−1r, we obtain D /∈ 〈Xs−1r〉H and ((r,C), (s,D)) /∈ 2R. Thus it only remains to consider the case d ∈ Xs−1r and θr−1c(d) 6= c. If θr−1s(d) ∈ C, as above we get ((s,d), (r,θr−1s(d))) /∈ R, which leads to a contradiction. This shows θr−1s(d) /∈ C, and ((r,C), (s,D)) /∈ 2R. � Combining [13, Lemma 34] with Lemma 3.6, Lemma 3.7 and using that the quotient map to the globalization is open we obtain the following. Theorem 3.8. Let θ be a partial action of G on X. Then XG is Hausdorff if and only if HG is Hausdorff. Recall that a locally compact Cantor space is a locally compact Hausdorff space with a countable basis of clopen sets and no isolated points. If a locally compact Cantor space X is compact, then there is a homeomorphism between X and the Cantor space. We proceed with the next. Proposition 3.9. Let X be a metric compact Cantor space, G a countable discrete group and suppose that θ = (θg,Xg)g∈G is a partial action of G on X such that Xg is clopen for all g ∈ G. Then (2X)G is a locally compact Cantor space. Proof. Since X is a compact Hausdorff space, then 2X is a compact Hausdorff space. Moreover since X is metric, we get from [12, Proposition 8.4] that 2X has no isolated points, and from [12, Proposition 8.6] we have that 2X have a countable basis of clopen sets. Therefore H1 = 2X is the Cantor space. Also 〈Xg〉 is clopen for all g ∈ G and the result follows from [9, Proposition 2.3]. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 264 Partial actions of groups on hyperspaces Now we shall work with the hyperspace H3 = F(X) consisting of finite subsets of X. The following result shows that the enveloping space (H3)G is T1, provided that XG is. Proposition 3.10. Let θ be a topological partial action of G on X and 2θ be the induced partial action of G on H3. If XG is T1, then (H3)G is T1. Proof. Let A = {a1, · · · ,an} ∈ H3 and g ∈ G, and qH3 : G×H3 → (H3)G be the corresponding quotient map. We need to show that q−1H3 ([g,A]) = {(h,F) ∈ G×H3 : ∃(g −1h) ·F and (g−1h) ·F = A} is closed in G×H3. Take (h,F) /∈ q−1H3 ([g,A]). There are two cases to consider. Case 1: @(g−1h) ·F. Then there is f ∈ F such that @(g−1h) ·f. Since XG is T1, for 1 ≤ i ≤ n there are open sets Ui ⊆ G and Vi ⊆ X for which (h,f) ∈ Ui ×Vi ⊆ (G×X) \q−1([g,ai]). Take U := ⋂n i=1 Ui and V = ⋂n i=1 Vi. Note that (h,F) ∈ U ×〈X,V 〉H3 ⊆ (G ×H3) \ q−1H3 ([g,A]). Indeed, if (t,B) ∈ U ×〈X,V 〉H3 and b ∈ B ∩ V we have (t,b) /∈ q−1([g,ai]) for any 1 ≤ i ≤ n. If @g−1t · b, then @(g−1t) · B and (t,B) /∈ q−1H3 ([g,A]). On the other hand, if ∃(g −1t) · b, then (g−1t) · b 6= ai, for each 1 ≤ i ≤ n, then (g−1t) · b /∈ A and (t,B) /∈ q−1H3 ([g,A]). Case 2: ∃(g−1h) · F and (g−1h) · F 6= A. If there is f ∈ F for which (g−1h) · f /∈ A we get (h,f) /∈ q−1([g,ai]) for 1 ≤ i ≤ n and we proceed as in Case 1. If there is a ∈ A such that (g−1h) · f 6= a, for any f ∈ F write F = {f1, · · · ,fk}, then (h,fj) /∈ q−1([g,a]) for each 1 ≤ j ≤ k. Hence there are open sets U ⊆ G and V ⊆ X such that (h,fj) ∈ U ×V ⊆ (G×X)\q−1([g,a]), for every 1 ≤ j ≤ k. Note that (h,F) ∈ U ×〈V 〉H3 ⊆ (G×H3) \ q −1 H3 ([g,A]). Indeed, if (t,B) ∈ U ×〈V 〉H3 . If @(g−1t) ·B, then (t,B) /∈ q −1 H3 ([g,A]). In the case ∃g−1t ·B, we get that for any b ∈ B the pair (t,b) belongs to U ×V and thus (g−1t) · b 6= a which gives (t,B) /∈ q−1H3 ([g,A]), as desired. � Combining Lemma 3.6 and Proposition 3.10 we get. Corollary 3.11. Let θ be a topological partial action of G on X and 2θ be the induced partial action of G on H3. Then (H3)G is T1 if and only if XG is T1. We proceed with the next Corollary 3.12. Let G be a separable group and θ be a continuous partial action of G on X such that G ∗ X is open and X is separable. Take H ∈ {H1,H3} then the following assertions hold. (i) XG is separable; (ii) HG is separable; (iii) If XG is T1, then H(XG) and H(F(X)G) are separable. Proof. (i) Since θ is continuous with open domain then ι(X) is open in XG and (2.4) is a homeomorphism onto ι(X), in particular ι(X) is separable, moreover © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 265 J. L. Mart́ınez , H. Pinedo and E. Ramirez the map µ given in (2.3) acts continuously in XG and G · ι(X) = XG thus the result follows by (i) in Lemma (2.4). (ii) Since X is separable and T1, then H is separable. Then by (i) and (ii) of Theorem 3.2 and Lemma (2.4) we get that HG is separable. (iii) By (i) the space XG is separable and follows that H(XG) is separable. Finally, by Proposition 3.10 we have that F(X)G is T1, moreover F(X)G is separable thanks to (ii), and thus H(F(X)G) is separable. � Example 3.13 ([13, Example 4.8]). Consider the partial action of Z on X = [0, 1] given by θ0 = idX and θn = id[0,1),n 6= 0 then θ is continuous with open domain and XZ is T1. Thus by Corollary 3.12 the spaces XZ,HZ,H([0, 1]Z) and H(F([0, 1])G) are separable, where H∈{H1,H3}. Now we shall deal with the regularity condition. Theorem 3.14. Let θ : G∗X → X be a continuous partial action with clopen domain. Then the spaces XG and HG are regular, provided that H∈{H1,H2}. Proof. Let ι be the embedding map defined in (2.4) then Gι(X) = XG, we shall prove that ι(X) is clopen and regular. Let q : G × X → XG be the quotient map, then q−1(ι(X)) = G ∗ X is clopen in G × X which shows that ι(X) is clopen in XG. Now since X is a compact Hausdorff space we have that ι(X) is regular and thus XG is regular thanks to item (ii) of Lemma 2.4. On the other hand, we have that H is compact and Hausdorff, 2θ is continuous ((ii) of Theorem 2.8) and G∗H is clopen thanks to of Theorem 3.2 and Proposition 3.5 , then it is enough to apply (ii) of Lemma 2.4. � Example 3.15 ([8, p. 22]). Partial Bernoulli action Let G be a discrete group and X := {0, 1}G. The map β : G × X 3 (g,ω) 7→ gω ∈ X, is a continuous global action. The topological partial Bernoulli action is obtained by restricting β to the open set Ω1 = {ω ∈ X : ω(1) = 1} (see (2.1)). It is shown in [11, Example 3.4] that G ∗ Ω1 is clopen. Thus by Theorem 3.14 we have that HG is regular where H ∈ {H1,H2}. Moreover, since G · Ω1 = X, then XG = {0, 1}G is regular. Remark 3.16. In [13, Theorem 4.6] are presented other conditions for the space XG being regular. 4. On the category G y CH We shall use some of the above results to construct a monad in the category of partial actions on compact Hausdorff spaces. First recall the next. Definition 4.1. Let φ = (φg,Xg)g∈G and ψ = (ψg,Yg)g∈G, be partial actions of G on the spaces X and Y , respectively. A G-map f : φ → ψ is a continuous function f : X → Y such that: (i) f(Xg) ⊆ Yg, (ii) f(φg(x)) = ψg(f(x)), for each x ∈ Xg−1, © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 266 Partial actions of groups on hyperspaces for any g ∈ G. If moreover f is a homeomorphism and f−1 is G-map, we say that φ are ψ equivalent. We denote by G y Top the category whose objects are topological partial actions of G on topological spaces and morphisms are G-maps defined as above. Also, we denote by G y CH the subcategory of G y Top whose objects are topological partial actions of G on compact Hausdorff spaces. It follows by Theorem 3.2 that there is a functor 2− : G y CH → G y CH. 4.1. The monad I. Recall the next. Definition 4.2. Let C be a category. A monad in C is a triple (T,η,µ), where T : C → C is an endofunctor, η : IdC =⇒ T and µ : T 2 =⇒ T are natural transformations such that: (4.1) µ◦Tη = µ◦ηT = 1T and µ◦µT = µ◦Tµ. Given an object α = (αg,Xg)g∈G ∈ G y CH. We have by Lemma 2.5 that the map ηα : X 3 x 7→ {x}∈ 2X, is a continuous function. From this it is not difficult to see that ηα : α → 2α is a morphism in G y CH. Moreover, for another object β = (Yg,βg)g∈G in G y CH and a morphism f : α → β the diagram: X f �� ηα // 2X 2f �� Y ηβ // 2Y is commutative. Thus the family η = {ηα}α∈GyCMet : IdGyCMet =⇒ 2− is a natural transformation. Now set µα : 2 2X 3 A 7→ ∪A ∈ 2X, by Lemma 2.5 µα is continuous. We shall check that µα : 2 2α → 2α is a morphism in G y CH . (i) Take g ∈ G and A ∈ 〈〈Xg〉〉. Then A ⊆ 〈Xg〉 and µα(A) = ∪A ⊆ Xg, that is µα(A) ∈ 〈Xg〉. (ii) For A ∈ 〈〈Xg−1〉〉 we have 2αg [A] = {αg(F) : F ∈ A}, then µα(22 αg (A)) = µα(2 αg [A]) = ∪2αg [A] = αg(∪A) = 2αg (∪A) = 2αg (µα(A)), as desired. Now we prove that µ = {µα}{α∈GyCH} : (2−)2 =⇒ 2− is a natural transfor- mation. For this take β = (Yg,βg)g∈G in G y CH and a morphism f : α → β in G y CH. Consider the diagram (4.2) 22 X 22 f �� µα // 2X 2f �� 22 Y µβ // 2Y Let A ∈ 22 X , then 2f [A] = {f(B) : B ∈ A} and 2f (µα(A)) = 2f (∪A) = f(∪A) = ∪2f [A] = µβ(2f [A]) thus the diagram (4.2) is commutative. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 267 J. L. Mart́ınez , H. Pinedo and E. Ramirez Theorem 4.3. Let η and µ be as above. Then the triple I = (2−,η,µ) forms a monad in the category G y CH. Proof. It remains to prove that equalities in (4.1) hold. Let α be an object in G y CH Since(η2−)α = η2α, we have that µα◦η2α : 2X 3 A 7→ A ∈ 2X, which gives µ◦η2− = 12− . Also, (2−η)α = 2−(ηα) = 2ηα, and µα ◦ 2ηα : 2X 3 A 7→ A ∈ 2X, which shows µ◦ 2−η− = µ◦η2− = 12− . Finally, since (µ2−)α = µ2α and (2−µ)α = 2 −(µα), we have (µ◦µ2−)α = µα ◦µ2α = µα ◦ 2µα = (µ◦ 2−µ)α, thus µ◦µ2− = µ◦ 2−µ and (2−,η,µ) is a monad. � Acknowledgements. 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