- Email: [email protected]
On metric orbit spaces and metric dimension
Topology and its Applications 214 (2016) 94–99
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On metric orbit spaces and metric dimension Majid Heydarpour Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 14 June 2016 Accepted 12 October 2016 Available online 14 October 2016 MSC: primary 54E35 secondary 51K05, 53C20
a b s t r a c t For a metric space (X, d), a subset A resolves (X, d) if each point x ∈ X is uniquely determined by the distances d(x, a) for a ∈ A. Also the metric dimension of (X, d) is the smallest cardinality md(X) such that there is a set A of the cardinality md(X) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds. © 2016 Elsevier B.V. All rights reserved.
Keywords: Metric orbit space Metric dimension Metric (X, G)-manifold Hyperbolic and spherical spaces Riemannian manifold Geometric space
1. Introduction In 1953, Blumental [2] for the first time introduced the concept of the metric dimension of a metric space. The concept received more attention by way of its application in the set of the vertices of a graph (e.g. [7,14]). Since then it has found further applications in many other disciplines (e.g. [3–5,10,12]). Bau and Beardon [1], returning to the original idea of the metric dimension of a metric space, computed among other things, the metric dimension for n-dimensional Euclidean space, spherical space, hyperbolic space, and Riemann surfaces. Recently, we [8] presented some generalizations of [1] and computed the metric dimensions of n-dimensional geometric spaces. (See [9] where the metric dimension for the metric manifolds has been computed.) In [6] using metric dimension, an interesting characterization of the points in a simplex for a normed space has been given. Let us recall from [1] that for a metric space (X, d) by a resolving set we mean a non-empty subset A of X with if d(x, a) = d(y, a) for all a ∈ A then x = y. The metric dimension md(X) of (X, d) is the smallest E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.10.004 0166-8641/© 2016 Elsevier B.V. All rights reserved.
M. Heydarpour / Topology and its Applications 214 (2016) 94–99
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cardinality κ such that there is a resolving subset of X with the cardinality κ. A subset of (X, d) with cardinality md(X) that resolves X is called a metric basis for X. As X resolves X every metric space X has a metric dimension which is at most the cardinality |X| of X. In this note our aim is to determine the metric dimension of a class of metric spaces called metric orbit spaces. A vast domain of topological spaces and topological manifolds fall in this class. This will give us a tool to compute the metric dimension for more general cases of metric spaces. 2. Preliminaries In this section we present some preliminary definitions. Our definitions and notation concerning topological and metric spaces and manifolds are standard; see, for example, [11]. As usual we define Euclidean space, hyperbolic space, and spherical space, respectively, by En = {x = (x1 , ..., xn ) | xi ∈ R} with the metric d(x, y) = x − y Hn = {x ∈ Rn | xn > 0} with the path metric derived from |dx|/xn Sn = {x ∈ Rn+1 | x = 1} with the path metric induced by the Euclidean metric on Rn+1 . We need the following fact from [1]. Lemma 2.1. Suppose X = En , Hn , Sn , or any open subset of En , then md(X) = n + 1. Let us remark that in a metric space X, the relation A ⊆ B ⊆ X does not, in general, imply neither md(A) ≤ md(B) nor md(B) ≤ md(A), see [8]. By an n-dimensional geometric space we mean a metric space (M, d) that is an n dimensional connected homogeneous Riemannian manifold. For example the spaces En , Hn , Sn , Tn (n-Torus), RPn (the real n-projective space), and CPn (the complex n-projective space) real Grassmannian O(n)/(O(r) × O(n − r)) and complex Grassmannian U (n)/(U (r) × U (n − r)) manifolds are elementary examples of geometric spaces. Let us remark that from [8] we know that for an n-dimensional geometric space X, md(X) = n + 1. For the main and equivalent definitions of a geometric space, see [13]. Definition 2.2. (i) Let G be a subgroup of S(X), the similarity group of an n-dimensional geometric space X and let M be an n-manifold. An (X, G)-atlas for M is defined as a family of charts Φ = {φi : Ui → X | i ∈ I} covering M such that the coordinate changes φj ◦ φ−1 : φi (Ui ∩ Uj ) −→ φj (Ui ∩ Uj ) i agree in a neighborhood of each point with an element of G. There is a unique maximal (X, G)-atlas for M containing Φ. An (X, G)-structure for M is a maximal (X, G)-atlas for M and an (X, G)-manifold is an n-manifold M together with an (X, G)-structure for M . (ii) A metric (X, G)-manifold is a connected (X, G)-manifold M such that G is a subgroup of I(X), the group of isometries of X.
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It can be shown that on a metric (X, G)-manifold M there exists a metric induced by X which is the path metric on M and generates a topology on M equivalent to its manifold topology. Throughout we consider this metric on a metric (X, G)-manifold M . We refer the reader to [13] for more details on X-metric on a metric (X, G)-manifold M and other related materials. For example let X = En , Hn , Sn . Then an (X, I(X))-structure on a manifold is called, respectively, a Euclidean, hyperbolic, and spherical structure and in this case we call it, respectively, a Euclidean, hyperbolic, and spherical n-manifold. We know from [9] that for an n-dimensional metric (X, G)-manifold M , md(M ) = n + 1. Now we are prepared to give the definition of main object in this note. Definition 2.3. (i) We say that the group G acts properly discontinuously on the topological space X, if G acts on X and for each compact subset K of X, the set K gK is nonempty for only finitely many g in G. Also we say that the action of G on X is free, if for each g in G and x in X, gx = x implies that g = e (e is the identity element of G). (ii) Let G be a group acting on the metric space X and the orbit space X/G be topologized with the quotient topology from X. Define dG : X/G × X/G → R by dG (Gx, Gy) = dist(Gx, Gy) = inf {d(x , y )|x ∈ Gx, y ∈ Gy} We know that if G acts properly discontinuously on X, then all of the G-orbits are closed subsets of X. Then in this case the distance function dG is a metric on orbit space X/G. As an example let Γ be a subgroup of the isometry group of H2 , the real hyperbolic plane, and acts properly discontinuously and freely on H2 (i.e. Γ is a Fuchsian group). Then H2 /Γ is a hyperbolic Riemann surface. Definition 2.4. An open subset F of a metric space X is a fundamental region for a group Γ of the isometries of X, if for each g and h in Γ, gF hF = ∅ and X = {gF | g ∈ Γ}. Finally let us remark that for a fundamental region F of a group Γ of the isometries of the metric space X acting properly discontinuously on X, there is a continuous bijection F¯ /Γ → X/Γ, and it is a homeomorphism if and only if F is locally finite [13, p. 237]. Also it is obvious that for each two points x and y in F , dΓ (Γx, Γy) = d(x, y). 3. Main results In this section we prove some results about the metric dimension of the metric orbit spaces in general and we also consider in some special cases. We start with an easy observation about metric orbit spaces in a special case. Lemma 3.1. Let X be a geometric space and Γ be a subgroup of the isometry group of X acting properly discontinuously and freely on X. Then md(X/Γ) = md(X). Proof. In this case X/Γ is a metric (X, I(X))-manifold [13] and by [9] the metric dimension of X/Γ is equal to the metric dimension of X. 2
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Example 3.1. In example, H2 /Γ is a metric (H2 , I(H2 ))-manifold, and by Lemma 3.1, md(H2 /Γ) = md(H2 ) = 3. We say that a metric space X is finitely compact if all its closed balls are compact. In the following lemma we give an upper bound for the metric dimension of metric orbit spaces for a special case. This shows clearly the role of the fundamental domain in the concept of the metric dimension. Lemma 3.2. Let X be a geodesically connected, geodesically complete and finitely compact metric space and let Γ be a group of isometries of X acting properly discontinuously on X. Then md(X/Γ) ≤ md(X). Proof. In the case that X is a geodesically connected, geodesically complete and finitely compact metric space, by [13, P.244], there is a locally finite fundamental domain F for Γ and we know that each open ball of X includes a metric basis [8]. Since a fundamental domain is an open set, F includes a metric basis, say B. Also for each x, y in F we have dΓ (Γx, Γy) = dX (x, y). Now if for some x and y in X and each b in B dΓ (Γx, Γb) = dΓ (Γy, Γb), then, by the definition of the fundamental domain, there exists x and y in F such that dX (x , b) = dΓ (Γx, Γb) = dΓ (Γy, Γb) = dX (y , b) = dX (y , b). Since B is a metric basis for X, x = y and thus Γx = Γy. This shows that the set ΓB = { Γb | b ∈ B } is a solver set for the metric orbit space X/Γ. It follows that md(X/Γ) ≤ md(X). 2 Example 3.2. (a) For X = En , Hn , Sn , and each subgroup Γ of isometry group of X acting properly discontinuously on X md(X/Γ) ≤ md(X) = n + 1. (b) Let X = E2 and Ωl be the reflection of E2 in the line l in E2 and let Γ be the group generated by Ω. ¯ 2. Γ is a group acting properly discontinuously on E2 . E2 /Γ is isometric to the closed upper half-plain U ¯ 2 , B = {(−1, 0), (1, 0)} is a metric basis. Then Also for U ¯ 2 ) = 2 < 3 = md(E2 ). md(E2 /Γ) = md(U (c) Let X be the union of x-axis and y-axis and let Γ = { 1, Φ, Ωx , Ωy }. Where Ωx and Ωy are the reflections of X in the x-axis and y-axis in X and Φ is the antipodal map (i.e. Φ(x) = −x). Since Γ is a finite group, then it acts properly discontinuously on X. But Γ has no fundamental region (every point of X is fixed by a nonidentity element of Γ). It is obvious that X/Γ is homeomorphic to R. Then md(X/Γ) = md(R) = 2 < 3 = md(X). Now we are ready to prove our main result, Theorem 3.3. Let X be a metric space and let Γ be a group of isometries of X acting properly discontinuously on X. Then md(X/Γ) ≤ md(X).
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Proof. Let B be a metric basis for X and let ΓB = { Γb | b ∈ B }. We show that ΓB is a metric basis for X/Γ. Suppose for some x and y in X and for each b in B dΓ (Γx, Γb) = dΓ (Γy, Γb). Since Γ acts properly discontinuously on X, each Γ orbit is closed, then there exists x in Γx, y in Γy, b , b in Γb and g, h in Γ such that b = gb and b = hb such that dΓ (Γx, Γb) = dX (x , b ) Since g and consequently g −1 are isometries of X, then dΓ (Γx, Γb) = dX (x , b ) = dX (x , gb) = dX (g −1 x , b) and similarly dΓ (Γy, Γb) = dX (y , b ) = dX (y , hb) = dX (h−1 y , b). It follows that for each b in B we have dX (g −1 x , b) = dX (h−1 y , b). Since B is a metric basis for X, x := g −1 x = h−1 y =: y . Then for some k in Γ, x = kx and x = g −1 kx and x is in Γx, and, similarly y is in Γy. Then Γx = Γx = Γy = Γy. This shows that ΓB is a solver for X/Γ. Now we conclude md(X/Γ) ≤ md(X). 2 We finish this note with the following examples showing that there is no lower bound in previous theorem. Example 3.3. Let X be the union of two perpendicular grate circles l = {x ∈ S 2 |x3 = 0} and m = {x ∈ S 2 |x2 = 0} in S 2 and let Ωl and Ωm be the limitations of the reflections of S 2 in the circles l and m to X and Φ be the limitation of the antipodal map of S 2 to X (i.e. Φ(x) = −x). (a) Let Γ = { 1, Φ}. Since Γ is a finite group, then it acts properly discontinuously on X. It is obvious that X/Γ is homeomorphic to S 1 ∧ S 1 . Then md(X/Γ) = md(S 1 ∧ S 1 ) = 3 = md(X). (b) Let Γ = { 1, Ωl , Ωm }. Since Γ is a finite group, then it acts properly discontinuously on X. Also we know that X/Γ is homeomorphic to S 1 . Then md(X/Γ) = md(S 1 ) = 2 < 3 = md(X). (c) Let Γ = { 1, Φ, Ωl , Ωm , ΦΩl , ΦΩm }. Since Γ is a finite group, then it acts properly discontinuously on X. X/Γ is homeomorphic to closed interval [0, 1]. Then md(X/Γ) = md([0, 1]) = 1 < 3 = md(X). Acknowledgement The author is grateful to Saeid Maghsoudi for reading of the first draft of the paper.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S. Bau, A.F. Beardon, The metric dimension of metric spaces, Comput. Methods Funct. Theory 13 (2013) 295–305. L.M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, 1953. D.L. Boutin, Determining sets, resolving sets, and the exchange property, Graphs Comb. 25 (2009) 789–806. J. Cáceres, C. Hernando, M. Mora, I.M. Pelayo, M.L. Puertas, On the metric dimension of infinite graphs, Electron. Notes Discrete Math. 35 (2009) 15–20. G.C. Chappell, J. Gimbel, C. Hartman, Bounds on the metric and partition dimension of a graph, Ars Comb. 88 (2008) 349–366. G.P. Gehér, Is it possible to determine a point lying in a simplex if we know the distances from the vertices?, J. Math. Anal. Appl. 439 (2016) 651–663. F. Harary, R.A. Melter, On the metric dimension of a graph, Ars Comb. 2 (1976) 191–195. M. Heydarpour, S. Maghsoudi, The metric dimension of geometric spaces, Topol. Appl. 178 (2014) 230–235. M. Heydarpour, S. Maghsoudi, The metric dimension of metric manifolds, Bull. Aust. Math. Soc. 91 (2015) 508–513. S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70 (1996) 217–229. J.M. Lee, Introduction to Smooth Manifolds, second ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. R.A. Melter, I. Tomescu, Metric basis in digital geometry, Comput. Vis. Graph. Image Process. 25 (1984) 113–121. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, New York, 1994. P.J. Slater, Leaves of trees, Congr. Numer. 14 (1975) 549–559.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On metric orbit spaces and metric dimension Majid Heydarpour Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 14 June 2016 Accepted 12 October 2016 Available online 14 October 2016 MSC: primary 54E35 secondary 51K05, 53C20
a b s t r a c t For a metric space (X, d), a subset A resolves (X, d) if each point x ∈ X is uniquely determined by the distances d(x, a) for a ∈ A. Also the metric dimension of (X, d) is the smallest cardinality md(X) such that there is a set A of the cardinality md(X) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds. © 2016 Elsevier B.V. All rights reserved.
Keywords: Metric orbit space Metric dimension Metric (X, G)-manifold Hyperbolic and spherical spaces Riemannian manifold Geometric space
1. Introduction In 1953, Blumental [2] for the first time introduced the concept of the metric dimension of a metric space. The concept received more attention by way of its application in the set of the vertices of a graph (e.g. [7,14]). Since then it has found further applications in many other disciplines (e.g. [3–5,10,12]). Bau and Beardon [1], returning to the original idea of the metric dimension of a metric space, computed among other things, the metric dimension for n-dimensional Euclidean space, spherical space, hyperbolic space, and Riemann surfaces. Recently, we [8] presented some generalizations of [1] and computed the metric dimensions of n-dimensional geometric spaces. (See [9] where the metric dimension for the metric manifolds has been computed.) In [6] using metric dimension, an interesting characterization of the points in a simplex for a normed space has been given. Let us recall from [1] that for a metric space (X, d) by a resolving set we mean a non-empty subset A of X with if d(x, a) = d(y, a) for all a ∈ A then x = y. The metric dimension md(X) of (X, d) is the smallest E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.10.004 0166-8641/© 2016 Elsevier B.V. All rights reserved.
M. Heydarpour / Topology and its Applications 214 (2016) 94–99
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cardinality κ such that there is a resolving subset of X with the cardinality κ. A subset of (X, d) with cardinality md(X) that resolves X is called a metric basis for X. As X resolves X every metric space X has a metric dimension which is at most the cardinality |X| of X. In this note our aim is to determine the metric dimension of a class of metric spaces called metric orbit spaces. A vast domain of topological spaces and topological manifolds fall in this class. This will give us a tool to compute the metric dimension for more general cases of metric spaces. 2. Preliminaries In this section we present some preliminary definitions. Our definitions and notation concerning topological and metric spaces and manifolds are standard; see, for example, [11]. As usual we define Euclidean space, hyperbolic space, and spherical space, respectively, by En = {x = (x1 , ..., xn ) | xi ∈ R} with the metric d(x, y) = x − y Hn = {x ∈ Rn | xn > 0} with the path metric derived from |dx|/xn Sn = {x ∈ Rn+1 | x = 1} with the path metric induced by the Euclidean metric on Rn+1 . We need the following fact from [1]. Lemma 2.1. Suppose X = En , Hn , Sn , or any open subset of En , then md(X) = n + 1. Let us remark that in a metric space X, the relation A ⊆ B ⊆ X does not, in general, imply neither md(A) ≤ md(B) nor md(B) ≤ md(A), see [8]. By an n-dimensional geometric space we mean a metric space (M, d) that is an n dimensional connected homogeneous Riemannian manifold. For example the spaces En , Hn , Sn , Tn (n-Torus), RPn (the real n-projective space), and CPn (the complex n-projective space) real Grassmannian O(n)/(O(r) × O(n − r)) and complex Grassmannian U (n)/(U (r) × U (n − r)) manifolds are elementary examples of geometric spaces. Let us remark that from [8] we know that for an n-dimensional geometric space X, md(X) = n + 1. For the main and equivalent definitions of a geometric space, see [13]. Definition 2.2. (i) Let G be a subgroup of S(X), the similarity group of an n-dimensional geometric space X and let M be an n-manifold. An (X, G)-atlas for M is defined as a family of charts Φ = {φi : Ui → X | i ∈ I} covering M such that the coordinate changes φj ◦ φ−1 : φi (Ui ∩ Uj ) −→ φj (Ui ∩ Uj ) i agree in a neighborhood of each point with an element of G. There is a unique maximal (X, G)-atlas for M containing Φ. An (X, G)-structure for M is a maximal (X, G)-atlas for M and an (X, G)-manifold is an n-manifold M together with an (X, G)-structure for M . (ii) A metric (X, G)-manifold is a connected (X, G)-manifold M such that G is a subgroup of I(X), the group of isometries of X.
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It can be shown that on a metric (X, G)-manifold M there exists a metric induced by X which is the path metric on M and generates a topology on M equivalent to its manifold topology. Throughout we consider this metric on a metric (X, G)-manifold M . We refer the reader to [13] for more details on X-metric on a metric (X, G)-manifold M and other related materials. For example let X = En , Hn , Sn . Then an (X, I(X))-structure on a manifold is called, respectively, a Euclidean, hyperbolic, and spherical structure and in this case we call it, respectively, a Euclidean, hyperbolic, and spherical n-manifold. We know from [9] that for an n-dimensional metric (X, G)-manifold M , md(M ) = n + 1. Now we are prepared to give the definition of main object in this note. Definition 2.3. (i) We say that the group G acts properly discontinuously on the topological space X, if G acts on X and for each compact subset K of X, the set K gK is nonempty for only finitely many g in G. Also we say that the action of G on X is free, if for each g in G and x in X, gx = x implies that g = e (e is the identity element of G). (ii) Let G be a group acting on the metric space X and the orbit space X/G be topologized with the quotient topology from X. Define dG : X/G × X/G → R by dG (Gx, Gy) = dist(Gx, Gy) = inf {d(x , y )|x ∈ Gx, y ∈ Gy} We know that if G acts properly discontinuously on X, then all of the G-orbits are closed subsets of X. Then in this case the distance function dG is a metric on orbit space X/G. As an example let Γ be a subgroup of the isometry group of H2 , the real hyperbolic plane, and acts properly discontinuously and freely on H2 (i.e. Γ is a Fuchsian group). Then H2 /Γ is a hyperbolic Riemann surface. Definition 2.4. An open subset F of a metric space X is a fundamental region for a group Γ of the isometries of X, if for each g and h in Γ, gF hF = ∅ and X = {gF | g ∈ Γ}. Finally let us remark that for a fundamental region F of a group Γ of the isometries of the metric space X acting properly discontinuously on X, there is a continuous bijection F¯ /Γ → X/Γ, and it is a homeomorphism if and only if F is locally finite [13, p. 237]. Also it is obvious that for each two points x and y in F , dΓ (Γx, Γy) = d(x, y). 3. Main results In this section we prove some results about the metric dimension of the metric orbit spaces in general and we also consider in some special cases. We start with an easy observation about metric orbit spaces in a special case. Lemma 3.1. Let X be a geometric space and Γ be a subgroup of the isometry group of X acting properly discontinuously and freely on X. Then md(X/Γ) = md(X). Proof. In this case X/Γ is a metric (X, I(X))-manifold [13] and by [9] the metric dimension of X/Γ is equal to the metric dimension of X. 2
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Example 3.1. In example, H2 /Γ is a metric (H2 , I(H2 ))-manifold, and by Lemma 3.1, md(H2 /Γ) = md(H2 ) = 3. We say that a metric space X is finitely compact if all its closed balls are compact. In the following lemma we give an upper bound for the metric dimension of metric orbit spaces for a special case. This shows clearly the role of the fundamental domain in the concept of the metric dimension. Lemma 3.2. Let X be a geodesically connected, geodesically complete and finitely compact metric space and let Γ be a group of isometries of X acting properly discontinuously on X. Then md(X/Γ) ≤ md(X). Proof. In the case that X is a geodesically connected, geodesically complete and finitely compact metric space, by [13, P.244], there is a locally finite fundamental domain F for Γ and we know that each open ball of X includes a metric basis [8]. Since a fundamental domain is an open set, F includes a metric basis, say B. Also for each x, y in F we have dΓ (Γx, Γy) = dX (x, y). Now if for some x and y in X and each b in B dΓ (Γx, Γb) = dΓ (Γy, Γb), then, by the definition of the fundamental domain, there exists x and y in F such that dX (x , b) = dΓ (Γx, Γb) = dΓ (Γy, Γb) = dX (y , b) = dX (y , b). Since B is a metric basis for X, x = y and thus Γx = Γy. This shows that the set ΓB = { Γb | b ∈ B } is a solver set for the metric orbit space X/Γ. It follows that md(X/Γ) ≤ md(X). 2 Example 3.2. (a) For X = En , Hn , Sn , and each subgroup Γ of isometry group of X acting properly discontinuously on X md(X/Γ) ≤ md(X) = n + 1. (b) Let X = E2 and Ωl be the reflection of E2 in the line l in E2 and let Γ be the group generated by Ω. ¯ 2. Γ is a group acting properly discontinuously on E2 . E2 /Γ is isometric to the closed upper half-plain U ¯ 2 , B = {(−1, 0), (1, 0)} is a metric basis. Then Also for U ¯ 2 ) = 2 < 3 = md(E2 ). md(E2 /Γ) = md(U (c) Let X be the union of x-axis and y-axis and let Γ = { 1, Φ, Ωx , Ωy }. Where Ωx and Ωy are the reflections of X in the x-axis and y-axis in X and Φ is the antipodal map (i.e. Φ(x) = −x). Since Γ is a finite group, then it acts properly discontinuously on X. But Γ has no fundamental region (every point of X is fixed by a nonidentity element of Γ). It is obvious that X/Γ is homeomorphic to R. Then md(X/Γ) = md(R) = 2 < 3 = md(X). Now we are ready to prove our main result, Theorem 3.3. Let X be a metric space and let Γ be a group of isometries of X acting properly discontinuously on X. Then md(X/Γ) ≤ md(X).
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Proof. Let B be a metric basis for X and let ΓB = { Γb | b ∈ B }. We show that ΓB is a metric basis for X/Γ. Suppose for some x and y in X and for each b in B dΓ (Γx, Γb) = dΓ (Γy, Γb). Since Γ acts properly discontinuously on X, each Γ orbit is closed, then there exists x in Γx, y in Γy, b , b in Γb and g, h in Γ such that b = gb and b = hb such that dΓ (Γx, Γb) = dX (x , b ) Since g and consequently g −1 are isometries of X, then dΓ (Γx, Γb) = dX (x , b ) = dX (x , gb) = dX (g −1 x , b) and similarly dΓ (Γy, Γb) = dX (y , b ) = dX (y , hb) = dX (h−1 y , b). It follows that for each b in B we have dX (g −1 x , b) = dX (h−1 y , b). Since B is a metric basis for X, x := g −1 x = h−1 y =: y . Then for some k in Γ, x = kx and x = g −1 kx and x is in Γx, and, similarly y is in Γy. Then Γx = Γx = Γy = Γy. This shows that ΓB is a solver for X/Γ. Now we conclude md(X/Γ) ≤ md(X). 2 We finish this note with the following examples showing that there is no lower bound in previous theorem. Example 3.3. Let X be the union of two perpendicular grate circles l = {x ∈ S 2 |x3 = 0} and m = {x ∈ S 2 |x2 = 0} in S 2 and let Ωl and Ωm be the limitations of the reflections of S 2 in the circles l and m to X and Φ be the limitation of the antipodal map of S 2 to X (i.e. Φ(x) = −x). (a) Let Γ = { 1, Φ}. Since Γ is a finite group, then it acts properly discontinuously on X. It is obvious that X/Γ is homeomorphic to S 1 ∧ S 1 . Then md(X/Γ) = md(S 1 ∧ S 1 ) = 3 = md(X). (b) Let Γ = { 1, Ωl , Ωm }. Since Γ is a finite group, then it acts properly discontinuously on X. Also we know that X/Γ is homeomorphic to S 1 . Then md(X/Γ) = md(S 1 ) = 2 < 3 = md(X). (c) Let Γ = { 1, Φ, Ωl , Ωm , ΦΩl , ΦΩm }. Since Γ is a finite group, then it acts properly discontinuously on X. X/Γ is homeomorphic to closed interval [0, 1]. Then md(X/Γ) = md([0, 1]) = 1 < 3 = md(X). Acknowledgement The author is grateful to Saeid Maghsoudi for reading of the first draft of the paper.
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