# Some Results on Pixley–Roy Hyperspaces

Authors: Ljubiša D. R. Kočinac; Luong Quoc Tuyen; Ong Van Tuyen
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we prove that if a spaceX has a point-countable cn-network, then the Pixley-Roy hyperspace PRX also has a point-countable cn-network. If X is a regular space with a point-countable ck-network, then so does the Pixley-Roy hyperspace PRX. Moreover, if X has a point-countable sp-network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace PR2X also has a point-countable sp-network (resp., strict Pytkeev network). On the other hand, we show that if the Pixley–Roy hyperspace PRX has a countable cn-network (resp., sp-network and strict Pytkeev network), then so does X. By these results, we obtain that if the Pixley–Roy hyperspace PRX is a cosmic space (resp., P0-space, strict P0-space, and stric P0-space), then so is X. Furthermore, the Pixley-Roy hyperspace PRnS2 is not a k-space for each n≥2.


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## Body

## 1. Introduction and Preliminaries

The topological and generalized metric properties on Pixley–Roy hyperspaces have been studied by many authors from different points of view, as shown in [1–14]. Especially, they studied the relation between a space X satisfying such a property and its Pixley-Roy hyperspaces satisfying the same property.In this paper, we study relationships between some generalized metric properties of a spaceX and generalized metric properties of Pixley–Roy hyperspaces over X. We prove that if X has a point-countable cn-network, then PRX also has a point-countable cn-network. If X is regular with a point-countable ck-network, then so does PRX. Furthermore, if PRX has a countable cn-network, then X also has a countable cn-network. Next, we study the notion of strict Pytkeev network which was introduced by T. Banakh in 2015 [15], and the notion of sp-network that was introduced by X. Liu, C. Liu, and S. Lin in 2019 [16] on Pixley–Roy hyperspaces and obtained the following results:(1)
IfX has a point-countable sp-network (resp., strict Pytkeev network), then so does PR2X(2)
IfPRX has a countable sp-network (resp., strict Pytkeev network), then so does XWe also obtain that ifX is regular and PRX is a cosmic space (resp., P0-space, strict P0-space, and stric P0-space), then X is also a cosmic space (resp., P0-space, strict P0-space, stric P0-space). Moreover, we show that there exists a space X with a countable sp-network, but PRX and PRnX with n≥3 do not have point-countable sp-networks.On the other hand, in [9], M. Sakai proved that for each n∈ℕ, PRnSω is sequential but PR3Sω is not Fréchet–Urysohn. Moreover, neither PR2X nor PRSω is a k-space. However, he does not know whether or not PRnS2 is a k-space for each n∈ℕ. By the abovementioned results, we show that PR1S2 is second-countable but PRnS2 is not a k-space for each n≥2.Throughout this paper, all spaces are assumed to be at leastT1. ℕ denotes the set of all positive integers, and the first infinite ordinal is denoted by ω. Moreover, if P is a family of subsets of a space X and G⊂X, then we set PG=P∈P:P∩G≠∅.The Pixley–Roy hyperspacePRX over a space X, defined by C. Pixley and P. Roy in [6], is the set of all nonempty finite subsets of X with the topology generated by the sets of the form,(1)F,V=G∈PRX:F⊂G⊂V,where F∈PRX, and V is an open subset in X containing F. It is known that PRX is always zero-dimensional and completely regular [3].For eachn∈ℕ, let(2)PRnX=F∈PRX:F≤n.Remark 1 (see [12]).
For eachn∈ℕ, PRnX is a closed subspace of PRX and, in particular, PR1X is a closed discrete subspace of PRX.By Remark1, we have the following remark.Remark 2.
EveryPRmX is a closed subspace of PRnX for each m,n∈ℕ with m<n.For eachF∈PRX and A⊂X, denote(3)F,A=H∈PRX:F⊂H⊂A.Moreover, ifA⊂PRX, then we denote the closure of A in PRX by ClA.Definition 1.
LetP be a family of subsets of a space X.(1)
P is a network [17] for X, if for any neighborhood U of a point x∈X, there exists a set P∈P such that x∈P⊂U(2)
P is a k-network [18] for X, if whenever K is a compact subset of an open set U in X, there exists a finite subfamily ℱ⊂P such that K⊂∪ℱ⊂U(3)
P is a cn-network [19] for X, if for any neighborhood U of a point x∈X, the set ∪P∈P:x∈P⊂U is a neighborhood of x(4)
P is a ck-network [19] for X, if for any neighborhood U of a point x∈X, there is a neighborhood Ux of x such that for each compact subset K⊂Ux, there exists a finite subfamily ℱ⊂P satisfying x∈∩ℱ and K⊂∪ℱ⊂U(5)
P is a cp-network [19] for X, if for each x∈X if either x is an isolated point of X and x∈P, or for each subset A of X with x∈A¯∖A and for each neighborhood U of x there exists a set P∈P such that P∩A is infinite and x∈P⊂U(6)
P is a cs∗-network [15] in X if for each x∈X, each neighborhood U of x and any sequence xn⊂X converging to x there is P∈P such that P⊂U and P contains infinitely many members of the sequence xn(7)
P is a Pytkeev network [20–22] (resp., strict Pytkeev network) [15] for X, if P is a network for X, and for each neighborhood U of a point x in X and each subset A of X accumulating at x, there exists a set P∈P such that P∩A is infinite and P⊂U (resp., x∈P⊂U)(8)
P is an sp-network [16] for X, if for each x∈U∩A¯ with U open and A subset in X, there is a set P∈P such that x∈P⊂U and x∈P∩A¯(9)
P is called point-countable [17], if the family P∈P:x∈P is countable for each x∈XRemark 3 (see [19]).
Bases⇒ck-networks ⇒cn-networks ⇒ networks.Remark 4 (see [16]).
(1)
In [15, p. 152], it was said that a subset A of a topological space X accumulates at a point x∈X if each neighborhood of x contains infinitely many points of the set A. It is obvious that for a T1-space X, a point x∈X is an accumulation point of a set A⊂X if and only if x∈A∖x¯.(2)
IfX is a T1-space, then P is a strict Pytkeev network if and only if it is a cp-network for X.(3)
Every base forX is an sp-network; every sp-network for X is a strict Pytkeev network (i.e., a cp-network), and the converse is also true for a Fréchet–Urysohn space; every strict Pytkeev network is a Pytkeev network.Definition 2.
LetX be a regular space.(1)
X is said to be a cosmic space [15], if X has a countable network(2)
X is said to be a (strict) P0-space [15], if X has a countable (strict) Pytkeev network(3)
X is said to be a stric P0-space [16], if X has a countable sp-networkRemark 5 (see [15, 19]).
(1)
A regular spaceX is cosmic if and only if X has a countable cn-network(2)
X is a strict P0-space if and only if X is a P0-spaceDefinition 3 (see [17]).
(1)
A spaceX is called a k-space if X is a Hausdorff space and X is an image of a locally compact space under a quotient mapping(2)
A spaceX is called a sequential space if a set A⊂X is closed if and only if together with any sequence it contains all its limits(3)
A spaceX is called a Fréchet–Urysohn space if for each A⊂X and each x∈A¯, there is a sequence in A converging to the point x in XRemark 6 (see [17]).
(1)
Every Fréchet–Urysohn space is a sequential space(2)
Every sequential Hausdorff space is ak-spaceDefinition 4 (see [16]).
Let(4)X=∞∪xn:n∈ℕ∪xnm:m,n∈ℕ,where every xn, xnm, and ∞ are different from each other. The set X endowed with the following topology is called the Arens space and denoted briefly as S2: each xnm is isolated; a basic neighborhood of xn has the form xn∪xnm:m>k for some k∈ℕ; a basic neighborhood of ∞ has the form ∞∪∪Vn:n≥k for some k∈ℕ, where each Vn is a neighborhood of xn.Definition 5 (see [16]).
A topological spaceX is called the sequential fan, which is denoted briefly as Sω if X is the quotient space by identifying all the limit points of ω many nontrivial convergent sequences.


## 2. Main Results

LetP be a family of subsets of a space X. If we put(5)P=F,∪ℱ:F∈PRXandℱis a finite subfamily ofPF,then P is a family of subsets of PRX.We prove now a few technical lemmas which will be used in the sequel.Lemma 1.
LetX be a space. If U is an open subset of PRX, then ∪U is an open subset of X.Proof.
Letx∈∪U. Then, there exists F∈U such that x∈F. Since F∈U, there exists V open in X such that F∈F,V⊂U. Moreover, since x∈V, we only need to prove that V⊂∪U. Given a point z∈V. Then, F⊂z∪F⊂V. This implies that(6)z∪F∈F,V⊂U.
Therefore,z∈z∪F⊂∪U. It shows that V⊂∪U. □Lemma 2.
LetX be a space. If C is a compact subset of PRX, then ∪C is a compact subset of X.Proof.
LetU be an open cover of ∪C in X. Then, for each F∈C, F⊂∪C⊂∪U. Thus, for each x∈F, there exists Ux∈U such that x∈Ux. Since F∈F,∪x∈FUx for each F∈C, we claim that(7)U=F,∪x∈FUx:F∈C,is an open cover of C in PRX. On the other hand, since C is compact, there exists a finite subfamily V of U such that C⊂∪V. This implies that there exist F1,…,Fm∈C such that(8)V=F1,∪x∈F1Ux,…,Fm,∪x∈FmUx.
PutV=Ux:x∈Fi,i≤m. Because each set Fi is finite, V is a finite subfamily of U. Hence, we only need to prove that ∪C⊂∪V. Take any point z∈∪C, then z∈A for some A∈C. This implies that there exists i≤m such that A∈Fi,∪x∈FiUx, hence A⊂∪x∈FiUx. Therefore, z∈Ux⊂∪V for some x∈Fi. □Lemma 3.
IfP is point-countable for X, then P is point-countable for PRX.Proof.
LetF∈PRX and W∈P such that F∈W. Then, there exist G∈PRX and a finite subfamily ℱ⊂PG such that W=G,∪ℱ and G⊂F⊂∪ℱ. Since ℱ⊂PG, we have ℱ⊂PF. This implies that(9)W∈P:F∈W⊂G,∪ℱ:G⊂F,ℱis a finite subfamily ofPF.
BecauseP is point-countable for X and the set F is finite, PF is countable and the set G:G⊂F is finite. Thus,(10)G,∪ℱ:G⊂F,ℱis a finite subfamily ofPF,is countable. Therefore, W∈P:F∈W is countable. This proves that P is point-countable for PRX.Lemma 4.
(1)
IfP is a cn-network for X, then P is a cn-network for PRX(2)
IfX is regular and P is a ck-network for X, then P is a ck-network for PRXProof.
LetF∈PRX and U be an open neighborhood of F in PRX. Then, there exists V open in X such that F∈F,V⊂U.(1)
LetP be a cn-network for X. Put(11)ℋ=P∈PF:P⊂V.Then,∪ℋ is a neighborhood of F in X. This implies that there exists W open in X such that F⊂W⊂∪ℋ. Thus, F∈F,W⊂F,∪ℋ. Moreover, we claim that(12)F,∪ℋ⊂∪W∈P:F∈W⊂U.Indeed, take anyH∈F,∪ℋ, then F⊂H⊂∪ℋ. Because the set H is finite, there is a finite subfamily G⊂ℋ such that H⊂∪G. On the other hand, since ∪G⊂∪ℋ⊂V, F,∪G⊂F,V⊂U. Furthermore, since G⊂ℋ⊂PF, we have(13)F,∪G∈W∈P:F∈W⊂U.BecauseH∈F,∪G, this implies that H∈∪W∈P:F∈W⊂U.Therefore,(14)F∈F,W⊂F,∪ℋ⊂∪W∈P:F∈W⊂U.Since the setF,W is open in PRX, ∪W∈P:F∈W⊂U is a neighborhood of F in PRX. This shows that P is a cn-network for PRX.(2)
Suppose thatP is a ck-network for X. Then, for each x∈F, there exists a neighborhood Vx of x such that Vx⊂V and for each compact subset Ax⊂Vx, there exists a finite subfamily Ax of P satisfying x∈∩Ax and Ax⊂∪Ax⊂V. For each x∈F, since X is regular, there is an open set Wx in X such that x∈Wx⊂W¯x⊂Vx. Put VF=F,∪x∈FWx, then for each compact subset K⊂VF, we have(15)∪K⊂∪x∈FW¯x.(i)
By Lemma2, ∪K is compact in X. Thus, Kx=∪K∩W¯x is compact in X and Kx⊂Vx for each x∈F. This implies that there is a finite subfamily ℱx⊂P such that x∈∩ℱx and Kx⊂∪ℱx⊂V. Put(16)ℋ=∪x∈Fℱxandℋ=∪x∈Fℱx,andF=F,∪G≠∅:G⊂ℋ.Then,F∈∩F and ∪F⊂F,V. On the other hand, since the set F is finite and ℋ⊂PF, the family F is a finite subfamily of P. Furthermore, we have K⊂∪F. In fact, take any H∈K, then H⊂∪K. For each y∈H, since ∪K=∪x∈FKx, there exists xy∈F such that y∈Kxy⊂∪ℱxy. Put G=∪y∈Hℱxy. Then, since H∈K⊂VF, F⊂H. This implies that H∈F,∪G∈F. Hence, H∈∪F. It shows that K⊂∪F. Thus, K⊂∪F⊂F,V⊂U. Therefore, P is a ck-network for PRX.
By Lemmas3 and 4, we obtain the following theorem.Theorem 1.
LetX be a space.(1)
IfX has a point-countable cn-network, then so does PRX(2)
IfX is a regular space with a point-countable ck-network, then so does PRXQuestion 1.
LetX be a space. If PRX has a point-countable cn-network (resp., ck-network), then does X have a point-countable cn-network (resp., ck-network)?Theorem 2.
LetX be a space. If PRX has a countable cn-network, then so does X.Proof.
Assume thatP is a countable cn-network for PRX. Put(17)P=∪W:W∈P.
It is clear thatP is countable. Let x∈U with U open in X. Then, x,U is an open neighborhood of x in PRX. Because P is a cn-network for PRX, we have that(18)∪W∈P:x∈W⊂x,U,is a neighborhood of x in PRX. This implies that there is an open set U in PRX such that(19)x∈U⊂∪W∈P:x∈W⊂x,U.
Therefore,(20)x∈∪U⊂∪∪W:x∈∪W⊂U.
Hence,∪∪W:x∈∪W⊂U is a neighborhood of x in X. It shows that P is a cn-network for X. □Question 2.
LetX be a space. If PRX has a countable ck-network, then does X have a countable ck-network?Remark 7.
LetX be a space. Then, PR1X has a point-countable base that is defined by(21)P1=x:x∈X.
In particular, ifX is countable, then P1 is a countable base for PR1X.Lemma 5.
LetX be a space and n∈ℕ. Then, F=x1,…,xn is not an accumulation point of any subset of PRnX.Proof.
Otherwise, suppose that there is a subsetA of PRnX that accumulates at F. Then, F∈ClA\F∩PRnX. This implies that(22)F,X∩A\F≠∅.
Hence, there existsG∈A\F such that F⊂G. Since F=x1,…,xn and G∈PRnX, we conclude that G=F, which is a contradiction.Theorem 3.
LetX be a space. If X has a point-countable sp-network (resp., strict Pytkeev network), then so does PR2X.Proof.
LetP be a point-countable sp-network (resp., strict Pytkeev network) for X. Put(23)G=F,∪ℱ∩PR2X:F∈PR2Xandℱis a finite subfamily ofPF.
Since the point-countable property is hereditary, the familyG is point-countable for PR2X by Lemma 3.(i)
P is an sp-network for X.Given a pointF∈U∩ClA∩PR2X with U open and A subset in PR2X. Then, there exists an open neighborhood V of F in X such that(24)F∈F,V∩PR2X⊂U.Case 1:F=x. In this case, if x∈A, then since P is a network for X and x∈V, there exists P∈P such that x∈P⊂V. Clearly, x,P∩PR2X∈G satisfies the conditions of an sp-network. Otherwise, we put(25)A=x∪y:y≠x,x,y∈A.Becausex∈ClA∩PR2X, x,V∩A≠∅. On the other hand, since x∉A, there exists y∈V∖x such that x,y∈A. This implies that A≠x. Take any open neighborhood U of x in X. Similarly, we can show that there is a point y∈U∖x such that y∈U∩A∖x. Therefore, x∈A∖x¯.Next, sincex∈V∩A∖x¯ and P is an sp-network for X, there is a set P∈P satisfying x∈P⊂V and x∈P∩A∖x¯.Then,x,P∩PR2X∈G and(26)x∈x,P∩PR2X⊂x,V∩PR2X⊂U.Furthermore, we claim that(27)x∈Clx,P∩A∩PR2X.Indeed, letW be any open neighborhood of x in PRX. Then, there exists an open neighborhood W of x in X such that x∈x,W⊂W. Because x∈P∩A∖x¯, we have W∩P∩A∖x≠∅. Thus, there exists z∈W∩P∩A∖x. This implies that(28)x,z∈x,W∩x,P∩x,A∖x.Therefore,x,z∈W∩x,P∩A, hence W∩x,P∩A≠∅. This shows that x∈Clx,P∩A∩PR2X.Case 2:F=x1,x2. In this case, it is easy to check that x1,x2∈A. Then, since x1,x2∈V and P is a network for X, there exist P1,P2∈P such that x1∈P1⊂V and x2∈P2⊂V. Observe that x1,x2,P1∪P2∩PR2X∈G satisfies the conditions of an sp-network.Therefore, we conclude thatG is an sp-network for PR2X.(2)
P is a strict Pytkeev network for X.
First, we will prove thatG is a network for PR2X. Indeed, assume that F∈PR2X and U is an open neighborhood of F in PR2X. Then, there is an open set V in X satisfying,(29)F∈F,V∩PR2X⊂U.
BecauseV is an open neighborhood of F in X and P is a network for X, for each x∈F, there is a set Px∈P such that x∈Px⊂V. This implies that(30)F∈F,∪x∈FPx∩PR2X⊂F,V∩PR2X⊂U.
Thus,G is a network for PR2X.
Next, by Lemma5, x1,x2∈PR2X is not an accumulation point of any subset A of PR2X. Thus, suppose that A⊂PR2X accumulates at x and U is an open neighborhood of x in PR2X. Then, x∈ClA\x∩PR2X, and there exists an open neighborhood V of x in X such that(31)x∈x,V∩PR2X⊂U.
This implies that(32)x,V∩A\x≠∅.
Hence, there is a pointy∈V∖x such that x,y∈A. Put(33)A=x∪y:y≠x,x,y∈A.
LetU be any open neighborhood of x in X. It is easy to see that there exists y∈U∩A∖x. Thus, x∈A∖x¯.
Finally, becauseP is a strict Pytkeev network for X, there exists a set P∈P such that P∩A is infinite and x∈P⊂V. Then, the set x,P∩PR2X∈P satisfies that(34)x,P∩PR2X∩x,A=x,P∩A∩PR2X,is infinite. This implies that x,P∩PR2X∩A is infinite. Furthermore,(35)x∈x,P∩PR2X⊂x,V∩PR2X⊂U.
Therefore,P is a strict Pytkeev network for PR2X.Lemma 6 (see [16, Theorem 4.2]).
The following conditions are equivalent for a Hausdorff spaceX.(1)
X is a k-space with a point-countable sp-network(2)
X is a Fréchet-Urysohn space with a point-countable cs∗-networkExample 1.
There exists a spaceX with a countable sp-network but PRnX with n≥3 and PRX do not have point-countable sp-networks.Proof.
It follows from [16, example 4.3] that the sequential fan Sω has a countable sp-network. By [9, Theorem 3.9] and Remark 6, PR3Sω is a k-space. However, PR3Sω does not have a point-countable sp-network. Otherwise, PR3Sω is a Fréchet–Urysohn space by Lemma 6. This is a contradiction, as shown in [9, p. 313]. It is easy to check that the point-countable sp-network property is hereditary. Therefore, PRnSω with n≥3 and PRSω do not have point-countable sp-networks.Question 3.
LetX be a space and n≥3. If X has a point-countable strict Pytkeev network, then does PRnX have a point-countable strict Pytkeev network?Question 4.
LetX be a space. If PRX has a point-countable sp-network (resp., strict Pytkeev network), then does X have a point-countable sp-network (resp., strict Pytkeev network)?Theorem 4.
LetX be a space. If PRX has a countable sp-network (resp., strict Pytkeev network), then so does X.Proof.
Suppose thatP is a countable sp-network (resp., strict Pytkeev network) for PRX. Put(36)P=∪W:W∈P,and observe that P is countable.(1)
P is an sp-network for PRX. Then, let x∈U∩A¯ with U open and A subset in X.Case 1.x∈A. Then, since P is a network for PRX, there exists W∈P such that x∈W⊂x,U. Moreover, we have x∈W∩x,A. Therefore,(37)x∈∪W⊂∪x,U=U,x∈∪W∩A⊂∪W∩A¯.Case 2.x∉A. First, we prove that(38)x∈Clx,A∪x∖x.Take any open neighborhoodV of x in PRX. Then, there is an open neighborhood V of x in X such that x∈x,V⊂V. Since x∈A¯, V∩A≠∅. On the other hand, because x∉A, there exists y≠x such that y∈V∩A. This implies that(39)x,y∈x,V∩x,A∪x∖x⊂V∩x,A∪x∖x.Hence, (38) is proved.Next, sincex∈x,U and by (1), we have(40)x∈x,U∩Clx,A∪x∖x.On the other hand, becauseP is an sp-network for PRX, there is a set W∈P satisfying(41)x∈W⊂x,U,x∈ClW∩x,A∪x∖x.Clearly,x∈∪W⊂U.Finally, we will prove thatx∈∪W∩A¯. Let W be an open neighborhood of x in X. Then, x,W is an open neighborhood of x in PRX. Since(42)x∈ClW∩x,A∪x∖x,we conclude that(43)x,W∩W∩x,A∪x∖x≠∅.Hence, there existM⊂W∩A and x∉M such that(44)x∪M∈x,W∩W∩x,A∪x∖x.This implies thatM⊂W∩∪W∩A. Thus, x∈∪W∩A¯. Therefore, P is an sp-network for X.(2)
P is a strict Pytkeev network for PRX.First, we will prove thatP is a network for X. Take any x∈U with U open in X. Then, x,U is an open neighborhood of x in PRX. Since P is a network for PRX, there exists W∈P such that x∈W⊂x,U. This implies that(45)x∈∪W⊂∪x,U=U.Thus,P is a network for X.Now, assume thatA⊂X accumulates at x and U is an open neighborhood of x in X.Case 1.x∈A. In this case, we show that(46)x∈Clx,A∖x.Indeed, take any open neighborhoodV of x in PRX. Then, there is an open neighborhood V of x in X satisfying x∈x,V⊂V. Since x∈A∖x¯, we have that V∩A∖x≠∅. Thus, there exists y≠x such that y∈V∩A. This implies that(47)x,y∈x,V∩x,A∖x⊂V∩x,A∖x.Hence, (2) is true.Becausex∈x,U and P is a strict Pytkeev network for PRX, by (46), there exists a set W∈P such that W∩x,A is infinite and x∈W⊂x,U. This implies that ∪W∩A is infinite and x∈∪W⊂U.Case 2.x∉A. Similar to the proof of Case 1, we claim that(48)x∈Clx,A∪x∖x.Thus, we also find a setW∈P such that W∩x,A∪x is infinite and x∈W⊂x,U. This shows that ∪W∩A is infinite and x∈∪W⊂U.
Therefore, we claim thatP is a strict Pytkeev network for X. □
By Definition2, Remark 5, Theorem 2, and Theorem 4, we have the following corollary.Corollary 1.
LetX be a regular space. Then, if PRX is a cosmic space (resp., P0-space, strict P0-space, stric P0-space), then so is X.Lemma 7.
(1)
Every point-countable Pytkeev network for a Hausdorff space is ak-network [18, Theorem 3.1].(2)
Everyk-space with a point-countable k-network is a sequential space [23, Corollary 3.4].Corollary 2.
For eachn≥2, PRnS2 is not a k-space.Proof.
Since each closed subspace of ak-space is a k-space and for each m<n, PRmS2 is closed in PRnS2 by Remark 2, we only need to prove that PR2S2 is not a k-space. Otherwise, PR2S2 is a k-space. By [16, example 3.7], the Arens space S2 is a regular sequential space with a countable strict Pytkeev network. It follows from Theorem 3 that PR2S2 has a point-countable strict Pytkeev network. By Remark 4 (3) and Lemma 7, we conclude that PR2S2 is a sequential space. This implies that PR2S2 is a Fréchet–Urysohn space by [9, Lemma 4.1].
BecausePR2S2 is a Fréchet–Urysohn space with a point-countable strict Pytkeev network, PR2S2 has a point-countable sp-network by Remark 4 (3). On the other hand, because PR2S2 is countable, it is separable. Furthermore, observe that PR2S2 is regular. It follows from [16, Theorem 4.11] that PR2S2 has a countable sp-network. Similar to the proof of Theorem 4, we claim that S2 has a countable sp-network, which is a contradiction, as shown in [16, example 3.7]. Therefore, PR2S2 is not a k-space.


## 3. Conclusion

In this paper, we demonstrated that a few topological properties of a spaceX are preserved by passing to the Pixley–Roy hyperspaces over X. On the other hand, a space X has some generalized metric properties whenever the Pixley–Roy hyperspace over X has such properties. In particular, there are nice relations between Pytkeev-type properties of a space X and its Pixley–Roy hyperspace PRX. We mention that these properties have been intensively studied in recent years.It would be interesting to do a similar research for the other known hyperspace topologies (for example for the Vietoris and Fell topologies on the spaces of closed subsets of a spaceX).



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*Source: 5878044-2022-03-07.xml*