@ Appl. Gen. Topol. 22, no. 2 (2021), 385-397doi:10.4995/agt.2021.15096 © AGT, UPV, 2021 Fixed point property of amenable planar vortexes James F. Peters a and Tane Vergili b a Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty of Arts and Sciences, Adiyaman University, 02040 Adiyaman, Turkey. (james.peters3@umanitoba.ca) b Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey. (tane.vergili@ktu.edu.tr) Communicated by I. Altun Abstract This article, dedicated to Mahlon M. Day, introduces free group presen- tations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer. 2010 MSC: 37C25; 55M20; 54E05; 55U10. Keywords: amenable group; CW space; fixed point; planar vortex; presen- tation. 1. Introduction This article introduces consequences of results for amenable groups and fixed points found by M.M. Day during the 1960s in terms of free group presenta- tions [17] of planar vortexes in a CW (Closure-finite Weak) space. Results given here spring from a fundamental result for fixed points given by L.E.J. Brouwer [3]. Theorem 1.1 (Brouwer Fixed Point Theorem [18, §4.7, p. 194]). Every continuous map from Rn to itself has a fixed point. Received 11 February 2021 – Accepted 20 May 2021 http://dx.doi.org/10.4995/agt.2021.15096 J. F. Peters and T. Vergili Briefly, let Σ be a finite group and let m(Σ) be a set of bounded, real-valued functions on Σ. Then Σ is amenable, provided there is a mean µ on m(Σ) which is both left and right invariant. Theorem 1.2 (Day Abelian Group [Semigroup] Theorem ([6, 7])). Every finite group is amenable. The study of amenable groups led to the following extension of the Kakutani- Markov Theorem by Day. Theorem 1.3 (Day Fixed Point Theorem ([7])). Let K be a compact convex subset of a locally convex linear topological space X, and let Σ be a semigroup (under functional composition) of continuous affine transformations of K into itself. If Σ, when regarded as an abstract semigroup, is amenable, or if it has a left-invariant mean, then there is in K a common fixed point of the family Σ. A direct consequence of Theorem 1.3 is that each amenable group of a planar vortex has a fixed point in a CW space. Definition 1.4. A planar vortex vorE is a finite cell complex, which is a collection of path-connected vertices in nested, filled 1-cycles in a CW complex K. A 1-cycle in vorE (denoted by cycA) is a sequence of edges with no end vertex and with a nonempty interior. A geometric realization of vorE is denoted by |vorE| on |K| in the Euclidean plane. A nonvoid collection of cell complexes K is a Closure finite Weak CW space, provided K is Hausdorff (every pair of distinct cells is contained in disjoint neighbourhoods [11, §5.1, p. 94]) and the collection of cell complexes in K sat- isfy the Whitehead [19, pp. 315-317], [20, §5, p. 223] CW conditions, namely, the closure of each cell complex is in K and the nonempty intersection of cell complexes is in K. A number of important results concerning fixed points in this paper spring from Čech proximities, leading to descriptive proximally continuous maps. A descriptive proximally continuous map is defined over descriptive Čech prox- imity spaces [5, §4.1] in which the description of a nonempty set is in the form of a feature vector derived from probe functions, one for each feature of the set. For the details, see App. C. 2. Conjugacy between proximal descriptively continuous maps This section introduces proximal conjugacy between two dynamical systems, which is an easy extension of topological conjugacy [1, §8.1,p. 243]. Proximal conjugacy is akin to strongly amenable groups in which each of its proximal topological actions has a fixed point [8]. Let ∑ denote either a semigroup or a group. Also, let lub, glb denote least upper bound and greatest lower bound, respectively, and let m( ∑ ) be the set of bounded, real-valued functions θ on ∑ for which ‖θ‖ = lubx∈ ∑ |θ(x)| . © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 386 Fixed point property of amenable planar vortexes A mean µ on m( ∑ ) is an element of the m( ∑ )∗ (in the conjugate space B∗ of a Banach space B [6, p.510]) such that, for each x ∈ m( ∑ ), we have glbx∈ ∑θ(x) ≤ µ(x) ≤ lubx∈ ∑θ(x). An element of µ of m( ∑ )∗ is left[right] invariant, provided µ(ℓσx) = µ(x) [µ(rσx) = µ(x)], for all x ∈ m( ∑ ), σ ∈ ∑ , where (ℓσx)σ ′ = x(σσ′) and (rσx)σ ′ = x(σ′σ) for all σ′ ∈ Σ. Definition 2.1 ([6]). A semigroup (also group) ∑ is amenable, provided there is a mean µ on m( ∑ ), which is both left and right invariant. Theorem 2.2. A free group presentation of a planar vortex in a CW space is amenable. Proof. This result is a direct consequence of Theorem 1.2 from Day [7] and (I) [6, p.516], since, by construction, every free group G presentation of a planar vortex in a CW space is finite and, by definition, an Abelian semigroup. � Theorem 2.2 stems from Day’s extension of Theorem 2.3 (restated by Day [7, p. 585]) to cover the case when the family in question is a semigroup. Theorem 2.3 (Kakutani-Markov Theorem [9, 10]). Let K be a compact con- vex set in a locally convex linear topological space, and let F be a commuting family of continuous, affine transformations, f, of K into itself. Then there is a common fixed point of the functions in F; that is, there is an x in K such that f(x) = x for every f in F. If a planar vortex vorE has no hole inside, then we no longer need to require it be convex. It is automatically convex and we have the following. Theorem 2.4. For a CW complex K, let (K, δ) be a proximity space that contains a planar vortex vorE without a planar hole and let f : vorE → vorE be proximal continuous. Then vorE has a fixed point of f. Proof. Since vorE is finite, the topology on the geometric realization |vorE| of vorE can be regarded as the subspace topology inherited from the Euclidean space R2. Then if we have the geometric realization of f, denoted |f|, we see that |f| is a continuous affine transformation. This is true since, f maps two near subsets to the two near subsets. Also notice that |vorE| is convex compact subset of R2 and the collection of maps {|fn| : n = 1, 2 . . .} is amenable, since it is a semigroup under composition. Then by Theorem 2.3, for the family of continuous affine transformations {|fn| : n = 1, 2, . . .}, there is an x in |vorE| such that f(x) = x and so |vorE| has a fixed point of |f|. This also allows us to conclude that vorE has a fixed point of f without considering the geometric realization. � If a planar vortex vorE has a (planar) hole inside, then we could consider a subset vorE such that its geometric realization is convex compact. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 387 J. F. Peters and T. Vergili Theorem 2.5. For a CW complex K, let (K, δ) be a proximity space that contains a planar vortex vorE with a planar hole and let X ⊂ vorE such that its geometric realization |X| is convex compact. If f : X → X is proximal continuous, then X has a fixed point under f. Proof. By the construction of a planar vortex, there is subset X of vorE such that its geometric realization |X| is a convex compact subset of |vorE|. Then a proximal continuous map f : X → X has a corresponding continuous affine transformation |f| : |X| → |X|. Again by Theorem 2.3, for the family of continuous affine transformations {|fn| : n = 1, 2, . . . }, there is an x in |X| such that f(x) = x and so |X| has a fixed point of |f|. Hence, vorE has a fixed point of f without considering the geometric realization. � Remark 2.6. From what have observed, notice that any vortex has a locally compact abelian group presentation, since it is a locally compact Hausdorff space and its underlying group structure is abelian. In that case, any proximal continuous map from a vortex to itself can be also considered as a group action. Hence, by a direct consequence of Theorem 2.2 and a result of a generalization of the Kakutani-Markov Theorem 2.3, each amenable vortex has a fixed point. Corollary 2.7. If f is a proximal continuous map from a vortex to itself, then f has a fixed point. Next, we introduce the (descriptive) proximal conjugate between two prox- imal (descriptive) continuous maps. Note that a (descriptive) C̆ech proximity space X together with a (descriptive) proximal continuous self map on X can be considered as a (descriptive) proximal dynamical system. Now we intro- duce a (descriptive) proximal conjugacy between two (descriptive) dynamical systems, so that the existence of it guarantees the (descriptive) dynamical sys- tems having equivalent flows and related (descriptive) fixed points. Definition 2.8. Two proximal continuous maps f : (X, δ1) → (X, δ1) and g : (Y, δ2) → (Y, δ2) are said to be proximal conjugates, provided there exists a proximal isomorphism h : (X, δ1) → (Y, δ2) such that g ◦ h = h ◦ f. The function h is called a proximal conjugacy between f and g. The following theorem states that if two proximal continuous maps are prox- imal conjugate, then their corresponding iterated functions are also proximal conjugate. Theorem 2.9. Let h be a proximal conjugacy between f : (X, δ1) → (X, δ1) and g : (Y, δ2) → (Y, δ2). Then for each A ⊆ X and n ∈ Z+, we have h(f n(A)) = gn(h(A)). Proof. The proof follows from the induction on n. � Definition 2.10. Let (X, δΦ) be a descriptive proximity space with a probe function Φ : X → Rn and A, B ∈ 2X. Then A and B are said to be descriptively equal, provided Φ(A) = Φ(B). In that case, we write A = des B. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 388 Fixed point property of amenable planar vortexes Definition 2.11. Let Φ(E) ∈ Rn be a vector of n real-values that describe a nonempty set E. Two proximal descriptive continuous maps f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2) are said to be proximal descriptive conju- gates, provided there exists a proximal descriptive isomorphism h : (X, δΦ1) → (Y, δΦ2) such that g ◦ h(A) = des h ◦ f(A) for any A ∈ 2X. The function h is called a proximal descriptive conjugacy between f and g. Remark 2.12. We see from the definition of a proximal descriptive conjugacy that g ◦ h(A) and h ◦ f(A) may not be equal but we have Φ2(g ◦ h(A)) = Φ2(h ◦ f(A)) for A ∈ 2X. Moreover g ◦ h(A) = des h◦ f(A) implies g(A) = des h◦ f ◦ h−1(A) and f(A) = des h−1 ◦ g ◦ h(A), so that we have the following commutative diagrams. Φ1(f(A)) = Φ1(h −1gh(A)) A f(A) h−1gh(A) h(A) gh(A) f h Φ1 Φ1 g h −1 h−1(C) fh−1(C) C g(C) hfh−1(C) Φ2(g(C)) = Φ2(hfh −1(C)) f hh −1 g Φ2 Φ2 Remark 2.13. For proximal descriptive conjugates f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2), Def. 2.11 tells us that for A ⊆ X and C ⊆ Y , we have Φ2(g ◦ h(A)) = Φ2(h ◦ f(A)), Φ1(f ◦ h −1(C)) = Φ1(h −1 ◦ g(C)). Note that if h is a proximal descriptive conjugacy between f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2), then A = des B implies h(A) = des h(B) for A, B ∈ 2X. Theorem 2.14. Let h be a proximal descriptive conjugacy between f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2). Then for each A ∈ 2 X and n ∈ Z+, we have h(fn(A)) = des gn(h(A)). Proof. The proof follows from the induction on n. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 389 J. F. Peters and T. Vergili Definition 2.15. Let (X, δΦ) be a descriptive proximity space with a probe function Φ : X → Rn, A ∈ 2X, and f : (X, δΦ) → (X, δΦ) a descriptive proximally continuous map. (i) A is a descriptive fixed subset of f, provided Φ(f(A)) = Φ(A). (ii) A is an eventual descriptive fixed subset of f, provided A is not a de- scriptive fixed subset while ft(A) is a descriptive fixed subset for some t 6= 1. (iii) A is an almost descriptive fixed subset of f, provided Φ(f(A)) = Φ(A) or Φ(f(A)) δΦ Φ(A) so that f(A) ∩ Φ A 6= ∅ by Lemma C.1. (For the analogy of a point being almost fixed in digital topology, we refer to [2, 16].) Corollary 2.16. Let h be a proximal descriptive conjugacy between f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2). a) If A is a descriptively fixed subset of f, then h(A) is a descriptively fixed subset of g. b) If A is an eventual descriptively fixed subset of f, then h(A) is an eventual descriptively fixed subset of g. c) If A is an almost descriptively fixed subset of f, then h(A) is an almost descriptively fixed subset of g. Proof. a) Let A be a descriptively fixed subset of f. That is, Φ1(f(A)) = Φ1(A). In other words, we have f(A) = des A. Since h is a proximal iso- morphism, h preserves desciptive proximity h(f(A)) = des h(A). By Theo- rem 2.14, g(h(A)) = des h(A) so that h(A) is a descriptively fixed subset of g. b) Let A be an eventual descriptively fixed subset of f. That is, A is not a descriptively fixed subset of f but Φ1(f n(A)) = Φ1(A) for some pos- itive integer n > 1. In other words, we have fn(A) = des A. Since h is a proximal isomorphism, h preserves being equal in a descriptive sense: h(fn(A)) = des h(A). By Theorem 2.14, gn(h(A)) = des h(A). Note that h(A) is not a descriptively fixed subset of g since A is not a descriptively fixed subset of f and h is an isomorphism. So, h(A) is an eventual descrip- tively fixed subset of g. c) Let A be an almost descriptively fixed subset of f. That is, f(A) = des A or A δΦ1 f(A). If f(A) = des A, then we are done. Let A δΦ1 f(A). Since h is a proximal isomorphism, we have h(A) δΦ2 h(f(A)). By Theorem 2.14, h(A) δΦ2 g(h(A)) so that h(A) is a descriptively fixed subset of g. � Further, the existence of proximal conjugacy between two dynamical sys- tems of cell complexes such as vortexes also guarantees isomorphic amenable © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 390 Fixed point property of amenable planar vortexes group structures and hence related fixed points, which is another consequence of Theorem 2.2. Corollary 2.17. If there exists a descriptive proximal conjugacy between two descriptive dynamical systems, then they have isomorphic descriptive fixed sub- sets. Proof. Let f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2) be proximal descriptive conjugates and h : (X, δΦ1) → (Y, δΦ2) be the proximal descriptive conjugacy between them. If A ∈ 2X is a descriptive fixed subset of f, then h(A) is a descriptive fixed subset of g by Corollary 2.16 so that A and h(A) are descriptively isomorphic. Similarly if B ∈ 2Y is a descriptive fixed subset of g, then h−1(B) is a descriptive fixed subset of f so that B and h−1(B) are descriptively isomorphic. Hence there is a one-to-one correspondence between the set of the descriptive fixed subsets of f and the set of the descriptive fixed subsets of g. � 3. Weak conjugacy between descriptive proximally continuous maps This section introduces weak conjugacy between descriptive proximally con- tinuous maps. Definition 3.1. Two proximally continuous maps f : (X, δ1) → (X, δ1) and g : (Y, δ2) → (Y, δ2) are said to be weakly proximal conjugates, provided there exists a proximal isomorphism h : (X, δ1) → (Y, δ2) such that for any A ∈ 2 X, g ◦ h(A) δ2 h ◦ f(A). Note that this also implies that f ◦ h −1(C) δ1 h −1 ◦ g(C) for any C ∈ 2Y . The function h is called a weakly proximal conjugacy between f and g. Theorem 3.2. Let h be a weakly proximal conjugacy between f : (X, δ1) → (X, δ1) and g : (Y, δ2) → (Y, δ2). Then for each A ∈ 2 X and n ∈ Z+, we have h(fn(A)) δ2 g n(h(A)). Proof. The proof follows from the induction on n. � Definition 3.3. Two descriptive proximally continuous maps f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2) are said to be weakly proximal descrip- tive conjugates, provided there exists a proximal descriptive isomorphism h : (X, δΦ1) → (Y, δΦ2) such that g ◦ h(A) δΦ2 h ◦ f(A) for any A ∈ 2 X. Note that this also implies f ◦ h−1(C) δΦ2 h −1 ◦ g(C) for any C ∈ 2Y . The function h is called a weakly proximal descriptive conjugacy between f and g. Remark 3.4. For weakly proximal descriptive conjugates f : (X, δΦ1) → (X, δΦ1) and g : (Y, δΦ2) → (Y, δΦ2), Def. 3.3 and Lemma C.1 tell us that for A ∈ 2 X and C ∈ 2Y , we have g ◦ h(A) ∩ Φ f ◦ h(A) 6= ∅, f ◦ h−1(C) ∩ Φ h −1 ◦ g(C) 6= ∅. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 391 J. F. Peters and T. Vergili Appendix A. Planar Vortexes This section briefly looks at planar vortex structures in planar CW spaces. For simplicity, we consider only 2 cycle vortexes containing a pair of nested 1-cycles that intersect or attached to each other via at least one bridge edge. Definition A.1. ([15]) Let cycA, cycB be a collection of path-connected ver- texes on nested filled 1-cycles (with cycB in the interior of cycA) defined on a finite, bounded, planar region in a CW space K. A planar 2 cycle vortex vorE is defined by vorE = cl(cycB) is contained (nested) in the interior of cl(cycA). ︷ ︸︸ ︷ {cl(cycA) : cl(cycB) ⊂ int(cl(cycA))} . A vortex containing adjacent non-intersecting cycles has a bridge edge at- tached to vertexes on the cycles. Definition A.2. A vortex bridge edge is an edge attached to vertexes on a pair of non-intersecting, filled 1-cycles. Remark A.3. From Def. A.1, the cycles in a 2 cycle vortex can either have nonempty intersection (see, e.g., cycA′ ∩ cycB′ 6= ∅ in |vorE′| in Fig. 1b) or there is a bridge edge between the cycles (see, e.g., >pq |vorE| in Fig. 1a). In effect, every pair of vertexes in a 2 cycle vortex is path-connected. Remark A.4. The structure of a 2 cycle vortex extends to a vortex with k > 2 nested filled 1-cycles, provided adjacent pairs of cycles cycA, cycA′ in a k-cycle vortex either intersect or there is a bridge edge attached between cycA, cycA′. (a) Vortex |vorE| with non-intersecting 1-cycles cycA, cycB (b) Vortex |vorE| with intersecting 1-cycles cycA ′ , cycB ′ Figure 1. Sample planar 2-cycle vortexes © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 392 Fixed point property of amenable planar vortexes Appendix B. Free Group Presentation of a Vortex A finite group G is free, provided every element x ∈ G is a linear combination of its basis elements (called generators). We write B to denote a nonempty set of generators { g1, . . . , g|B| } and G(B, +) to denote the free group with binary operation +. Example B.1. The basis {g1, g2, g3} generates a group G whose geometric realization is |vorE′| in Fig. 1b. The + operation on G corresponds to a move from a generator to a neighbouring vertex. For example, b = traversing 3 cycA′ & 3 cycB′ edges to reach b via g1, g2 ︷ ︸︸ ︷ 3g1 + 3g2 b = traversing 7 cycB′ edges to reach b via g1, g2 ︷ ︸︸ ︷ 4g1 + 3g2 b = traversing 1 cycB′ edge to reach b via g3 ︷ ︸︸ ︷ 0g1 + 1g3. The identity element 0 in G is represented by a zero move from a generator g to another vertex (denoted by 0g) and an inverse in G is represented by a reverse move −g. Definition B.2. Let 2K be the collection of cell complexes in a CW space K, vortex |vorE| ∈ 2K, basis B ∈ |vorE|, ki the i th integer coefficient in a linear combination ∑ i,j kigj of generating elements gj ∈ B. A free group G presentation of |vorE| is a continuous self-map f : 2K → 2K defined by f(|vorE|) =    v := ∑ i,j kigj : v ∈ |vorE| , gj ∈ B    = |vorE| 7→ free group G ︷ ︸︸ ︷ G( { g1, . . . , g|B| } , +). Appendix C. Descriptive Proximity Spaces This section briefly introduces descriptive Čech proximity spaces, paving the way for descriptive proximally continuous maps. The simplest form of proxim- ity relation (denoted by δ) on a nonempty set was introduced by E. Čech [4]. A nonempty set X equipped with the relation δ is a Čech proximity space (de- noted by (X, δ)), provided the following axioms are satisfied. Čech Axioms (P.0): All nonempty subsets in X are far from the empty set, i.e., A 6 δ ∅ for all A ⊆ X. (P.1): A δ B ⇒ B δ A. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 393 J. F. Peters and T. Vergili (P.2): A ∩ B 6= ∅ ⇒ A δ B. (P.3): A δ (B ∪ C) ⇒ A δ B or A δ C. Given that a nonempty set E has k ≥ 1 features such as Fermi energy EF e, cardinality Ecard, a description Φ(E) of E is a feature vector, i.e., Φ(E) = (EF e, Ecard). Nonempty sets A, B with overlapping descriptions are descrip- tively proximal (denoted by A δΦ B). The descriptive intersection of nonempty subsets in A ∪ B (denoted by A ∩ Φ B) is defined by A ∩ Φ B = i.e., Descriptions Φ(A) & Φ(B) overlap ︷ ︸︸ ︷ {x ∈ A ∪ B : Φ(x) ∈ Φ(A) ∩ Φ(B)} . Let 2X denote the collection of all subsets in a nonvoid set X. A nonempty set X equipped with the relation δΦ with non-void subsets A, B, C ∈ 2 X is a descriptive proximity space, provided the following descriptive forms of the Čech axioms are satisfied. Descriptive Čech Axioms (dP.0): All nonempty subsets in 2X are descriptively far from the empty set, i.e., A 6 δΦ ∅ for all A ∈ 2 X. (dP.1): A δΦ B ⇒ B δΦ A. (dP.2): A ∩ Φ B 6= ∅ ⇒ A δΦ B. (dP.3): A δΦ (B ∪ C) ⇒ A δΦ B or A δΦ C. The converse of Axiom (dp.2) also holds. Lemma C.1 ([14]). Let X be equipped with the relation δΦ, A, B ∈ 2 X. Then A δΦ B implies A ∩ Φ B 6= ∅. Proof. Let A, B ∈ 2X. By definition, A δΦ B implies that there is at least one member x ∈ A and y ∈ B so that Φ(x) = Φ(y), i.e., x and y have the same description. Then x, y ∈ A ∩ Φ B. Hence, A ∩ Φ B 6= ∅, which is the converse of (dp.2). � Theorem C.2. Let K be a cell complex, V or(K) ⊂ K a collection of planar vortexes equipped with the proximity δΦ and let vorA, vorB ∈ V or(K). Then vorA δΦ vorB implies vorA ∩ Φ vorB 6= ∅. Proof. Immediate from Lemma C.1. � Let (X, δ1) and (Y, δ2) be two Čech proximity spaces. Then a map f : (X, δ1) → (Y, δ2) is proximal continuous, provided A δ1 B implies f(A) δ2 f(B), i.e., f(A) δ2 f(B), provided f(A) ∩ f(B) 6= ∅ for A, B ∈ 2 X [12, §1.4]. In gen- eral, a proximal continuous function preserves the nearness of pairs of sets [11, © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 394 Fixed point property of amenable planar vortexes §1.7,p. 16]. Further, f is a proximal isomorphism, provided f is proximal con- tinuous with a proximal continuous inverse f−1. Let (X, δΦ1) and (Y, δΦ2) be descriptive proximity spaces with probe func- tions Φ1 : X → R n, Φ2 : Y → R n, and A, B ∈ 2X. Then a map f : (X, δΦ1) → (Y, δΦ2) is said to be descriptive proximally continuous, provided A δΦ1 B im- plies f(A) δΦ2 f(B), i.e., f(A) δΦ2 f(B), provided f(A) ∩ Φ f(B) 6= ∅. Further f is a descriptive proximal isomorphism, provided f and its inverse f−1 are descriptively proximally continuous. Definition C.3. Let (X, δΦ) be a descriptive Čech proximity space and f : (X, δΦ) → (X, δΦ) a descriptive proximally continuous map. A set A ∈ 2 X is said to be descriptively invariant with respect to f, provided Φ(f(A)) ⊆ Φ(A). Notice that if A is a descriptively invariant set with respect to f, then Φ(fn(A)) ⊆ Φ(A) for all positive integer n. Theorem C.4. Let (X, δΦ) be a descriptive Čech proximity space and f : (X, δΦ) → (X, δΦ) a proximal descriptive continuous map. If {Ai}i∈I ⊆ 2 X is a collection of descriptively invariant sets with respect to f, then i) ∪i∈IAi is descriptively invariant with respect to f, and ii) ∩i∈IAi is descriptively invariant with respect to f. Proof. From our assumption, we have Φ(f(Ai)) ⊆ Φ(Ai) for all i ∈ I so that i) f(∪i∈IAi) = ∪i∈If(Ai) Φ(f(∪i∈IAi)) = Φ(∪i∈If(Ai)) = ∪i∈IΦ(f(Ai)) ⊆ ∪i∈IΦ(Ai) and ii) f(∩i∈IAi) ⊆ ∩i∈If(Ai) Φ(f(∩i∈IAi)) ⊆ Φ(∩i∈If(Ai)) ⊆ ∩i∈IΦ(f(Ai)) ⊆ ∩i∈IΦ(Ai) � Theorem C.5. Let (X, δΦ) be a descriptive Čech proximity space and f : (X, δΦ) → (X, δΦ) a descriptive proximally continuous map. If A ∈ 2 X is de- scriptively invariant with respect to f then clδΦA is also descriptively invariant with respect to with respect to f. Proof. The descriptive closure of a subset A of X is defined in [13, §1.21.2] as follows: clΦA = {x ∈ X | x δΦ A}. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 395 J. F. Peters and T. Vergili Take an element x in clΦA so that x δΦ A and Φ(x) ∈ Φ(A) by Lemma C.1. Since f is descriptive proximally continuous f(x) δΦ f(A) and Φ(f(x)) ∈ Φ(f(A)) by Lemma C.1. We also have Φ(f(x)) ∈ Φ(A) since A is an invariant set with respect to f. Therefore f(x) δΦ A and f(x) ∈ clΦA. 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