Here, we investigate properties of plastic metric spaces. Shortly speaking, a metric space is plastic if every non-expansive bijection is an isometry, cf. Section 2.

We will observe that the plasticity property consists of a geometrical sub-problem and a topological sub-problem. That is the reason why plasticity of a metric space, which can be easily defined, evolves as a challenging mathematical problem. In particular, we observe that the plasticity of a metric space is not inherited from sup-spaces, i. e., from including spaces, and it does not inherit to sub-spaces, i. e., to included spaces.

In this study, we concentrate on metric spaces which are sub-spaces of the real axis, and in this apparently simple situation, the typical difficulties come to the light.

The probably first study devoted to the plasticity problem is the study mentioned in the reference [1]; however, the term "plasticity" appeared much later and the problem itself remained unnoticed for several decades. A short literature survey and the information about the current progress in solution of the problem are shown in Section 2.2.

The study is organized as follows. Section 2 introduces the basic concepts and illustrates the existence of non-expansive bijections in the case that the metric space is a union of closed intervals. This case demonstrates the geometrical aspects of the problem. Then, Section 3 discusses the plasticity of metric spaces by means of metric spaces which are unbounded sequences of points, investigates the relevance of accumulation points and continuous subsets, and attacks the more topological parts of the plasticity concept. Finally, Section 4 resumes the observations and gives a short outlook to further research.

We denote a metric space by (A, d) where A is the set of points and d : A × A → ℝ₊ = {x ∈ ℝ: x ≥ 0} is the distance obeying the known axioms of positivity, symmetry, non-degeneracy, and the triangle inequality.

Let us start with some interesting observations on simple situations of A, e. g., some sets of singular points.

Proposition 3.1. Let A={ai}i=-∞+∞⊂ℝ be an increasing sequence that obeys

and

Then (A, d) is not plastic.

Proof. The shift φ: a_i ↦ a_i−1 is an example of a non-expansive bijection which is not an isometry.

Remark 3.2. The relation sign in Equations (2), (3) might be commonly inverted so that the distances between two subsequent points of A decrease instead of increase, and the statement remains unchanged.

Furthermore, let us consider sets which are bounded from one side. Let us recall the definition of an accumulation point, which we will use in what follows.

Definition 3.3. An accumulation point (or limit point) of a set A in a metric space X is a point x, such that every neighborhood of x with respect to the metric on X contains a point of A which differs from the point x.

An accumulation point of a set A does not have to be an element of A. We will proceed with the following lemma.

Lemma 3.4. Let A ⊂ ℝ be a set without accumulation points which is bounded from one side. Let a be a minimal—or maximal—element of A and φ: A → A be a bijective non-expansive map. Then φ(a) = a.

Proof. Without loss of generality, we may consider the case when a is a minimal element. Assume φ(a) ≠ a. Then there is b ∈ A such that φ(b) = a.

Claim: Let be c ∈ A. Then c ≤ b implies φⁿ(c) ≤ b for every n ∈ ℕ.

Proof of the Claim: We will use the induction in n. Indeed, if φⁿ(c) ≤ b and φⁿ⁺¹(c) > b we have

This contradiction completes the proof of the Claim.

Since

we have φ(a) ≤ b. Thus, the Claim provides φⁿ(a) ≤ b for every n ∈ ℕ. Now, the segment [a, b] is a trap for those points, which were mapped there. Our aim is to find such a “trapped” point out of the interior of the segment [a, b] and show that this leads to a contradiction. There are only two possible cases.

Case 1: φ(a) = b. In this case, points a and b were swapped by φ. Then, such a “trapped” point is the closest from the right-hand side point to b. There is c > b such that d(b, c) < d(b, d) for any d > b. Such point c exists since A is unbounded from above and there is no accumulation points. The point c cannot be mapped outside the segment [a, b] since it gives the contradiction with non-expansiveness of φ.

Case 2: φ(a) < b. With such a condition, a “trapped” point is φ(a) itself.

In both cases, we have a point t which does not belong to the interior of the segment [a, b] such that φ(t) belongs to this interior. Consider an orbit of this point t, i. e., the set {φn(t)}n=1∞. Due to the bijectivity of φ, this orbit does not have repeating elements. Thus, we have obtained a bounded infinite subset in A which contradicts the fact that A does not have accumulation points.

Remark 3.5. The condition about the absence of accumulation points in Lemma 3.4 cannot be omitted.

This remark is confirmed by the following example.

Example 3.6. Let A = ℤ₊ ∪ Q, where Q={14+1n,n≥4}. The bijective non-expansive map φ is

We observe that φ is bijective and it does not save the minimal element of A. We check that it is non-expansive.

The described set is shown on the left of Figure 2.

Lemma 3.4 immediately implies the following corollary.

Corollary 3.7. Let A ⊂ ℝ be an unbounded set without accumulation points. Let A have a minimal or maximal element and let φ: A → A be a bijective non-expansive map. Then, φ is an isometry, moreover, the identity.

Proof. Without loss of generality, we may consider the case when a is a minimal element. Let us show that φ(x) = x for every x ∈ A. Indeed, for the minimal element a, Lemma 3.4 ensures that φ(a) = a. Now suppose for some fixed y ∈ A, the condition φ(x) = x holds for every x < y, x ∈ A. Consider

Then, φ|_A1: A₁ → A₁ is a bijective non-expansive map, and y is a minimal element. Then φ(y) = y due to Lemma 3.4.

Proposition 4.1 in Naimpally et al. [4] states that for convex (in the sense of the same study) metric spaces, hereditarily EC-plasticity implies boundedness. Moreover, for convex subsets in Euclidean ℝⁿ, hereditarily EC-plasticity and boundedness are equivalent. However, the authors note that convexity is a too strong condition.

In Naimpally et al. [4], Theorem 4.3 states that an unbounded metric space with at least one accumulation point contains a non-plastic subspace. Corollary 3.7 demonstrates that the presence of an accumulation point is essential in the mentioned theorem, since it allows to build examples of unbounded hereditarily plastic spaces.

Let us go back to Example 3.6 and remark another interesting property of non-expansive bijections on ℝ. Suppose we have a set A ⊂ ℝ and a function φ: A → A. We will say that φ preserves the relation “between” on the set A if for any x, y, z ∈ A with x < y < z we have φ(x) < φ(y) < φ(z). Example 3.6 shows that non-expansive bijections do not have to preserve the relation “between.” Surprisingly, there is an example demonstrating the same property with a set without any accumulation points.

Example 3.8. Let A = ℕ ∪ Q, where Q = {2k, k ∈ ℤ₋}. The bijective non-expansive map φ is defined by

The map φ does not preserve the relation “between” since −4 < −2 < 0 but φ(−2) < φ(−4) < φ(0). Let us check that φ is non-expansive.

The described set is shown on the right of Figure 2.

Furthermore, we are going to present a sufficient condition for a set in ℝ to be plastic. Let us introduce the set

Obviously, several pairs of points may be situated in the same distance. That is why for every p ∈ D_A, we call its multiplicity the number of pairs of points in A which are on the distance p. This multiplicity may be finite or infinite.

Theorem 3.9. Let A ⊂ ℝ has no accumulation points and let D_A has a maximal element of finite multiplicity or a minimal element of finite multiplicity. Then, (A, d) is a plastic metric space.

Proof. Without loss of generality, we may assume that D_A has a minimal element a ∈ ℝ of finite multiplicity k ∈ ℕ. Let us denote

Let us take x_i ≤ x_j for all i, j with 1 ≤ i < j ≤ 2k. Consider an arbitrary non-expansive bijection φ: A → A. Due to the non-expansiveness of φ, we may conclude that φ maps X_a bijectively onto itself. Thus, φ|_Xa is an isometry on X_a. In particular, we find d(x₁, x_2k) = d(φ(x₁), φ(x_2k)). Since this distance is the biggest one on X_a, either φ(x₁) = x₁ and φ(x_2k) = x_2k or φ(x₁) = x_2k and φ(x_2k) = x₁. We will refer them as cases 1 and 2, respectively. In the first case, obviously, for every x ∈ A with x₁ < x < x_2k, we get φ(x) = x, so, in this case, φ|_{[x1, x2k]∩A} is the identity. In the second case, if the structure of A allows it, φ|_{[x1, x2k]∩A} is the inversion, called total symmetry. Furthermore, following the similar procedure as in Lemma 3.4, we have that in the first case, φ is the identity, and in the second case φ is the total symmetry.

Remark 3.10. The conditions of Theorem 3.9 are sufficient but not necessary for the plasticity of a set without accumulation points.

To make sure that the previous Remark 3.10 is true, one may consider the space (ℤ, d). For D_ℤ, the minimal and the maximal elements are equal to 1 and have infinite multiplicity, but the space is plastic. However, we constructed the next example, which is less trivial, to show that plastic spaces which do not satisfy the condition of the previous theorem may have richer structure.

Example 3.11. Let A={ai}i=-∞i=∞⊂ℝ, where {ai}i=-∞i=∞ is an increasing sequence such that

The corresponding D_A has no minimal or maximal element. However, (A, d) is plastic. In fact, let φ: A → A be a non-expansive bijection. Then,

Suppose d(φ(a0),φ(a1))=1n, where n ≥ 2. Consider the open ball with the radius n − 1 centered in φ(a₀). Due to the structure of A, this ball contains only the point φ(a₁), except for the center. On the other hand, the open ball with the radius n − 1 centered in a₀, and for n ≥ 3, it contains more than two points, and for n = 2, it contains two points but does not contain a₁. In both cases, we have a contradiction to the non-expansiveness of the map φ. That is why the only possible option is as follows:

Furthermore, just in the same way as in Theorem 3.9, we have that φ is either the identity or the inversion.

Now let us speak about the subsets which contain a continuous part. One may prove the following statement in the same way as the Proposition 3.1.

Proposition 3.12. Let be

where b_i < a_i+1 be such a sequence of intervals that

and

for all i ∈ ℤ. Furthermore, there exists j ∈ ℤ such that

Then, (A, d) is not plastic.

Remark 3.13. In the same way as in Proposition 3.1, the relation signs in Equations (4-6) might be commonly inverted.

Here is one more observation.

Proposition 3.14. Let A ⊂ ℝ contain an interval (a, +∞) or (−∞, a). Then, (A, d) is not plastic.

Proof. Without loss of generality, we discuss the case with (a, +∞). Let us define the map φ with

This map is non-expansive, bijective, and, at the same time, not an isometry.

In Naimpally et al. [4], Theorem 3.9 shows the plasticity of the space ℝ\ℤ. Unfortunately, the proof misses the case that the non-expansive bijection is a symmetry. However, the statement itself is true. One may use the same reasoning to prove the next proposition.

Proposition 3.15. Let

where

Then, (A, d) is plastic.

Remark 3.16. Propositions 3.12 and 3.15 hold true with the closed intervals.

Remark 3.17. On the other hand, if we consider in the statement of Proposition 3.15 half-intervals,

(A, d) is already a non-plastic space.

Remark 3.18. If we consider in the same statement the set of the form,

(A, d) is also a non-plastic space.

Figure 3 illustrates the previous remark.

The reader easily provides more examples which consist of open or closed intervals together with half-intervals, all with the same lengths. Again, we remark that the end-points of the intervals are critical points for the plasticity property.

The analysis of plastic sub-spaces A of the real-line ℝ has shown that first, the Lipschitz continuity of the map φ: A → A with Lipschitz constant 1 leads to useful and instructive illustrations of the non-expansivity of the map φ, to which it is identical.

The plasticity property of a metric space turned out to contain two complementary aspects, a purely geometrical one and a topological one. Already on the real-line ℝ, the different nature of both aspects become visible. Whereas the geometrical aspect is an extension of the non-expansivity of φ on a simply connected interval, the topological aspect leads to the question whether two or more sub-intervals can be glued at critical points by piecewise translations. Therefore, the investigation of sub-spaces of the real-line ℝ gives an appropriate framework for the investigation of the plasticity of metric spaces.

We expect that the interplay between the two types of nature of the problem gets more severe in higher dimensions. Already unions of rectangles and cuboids as sub-spaces of the d-dimensional Euclidean space ℝ^d give a tremendous multiplicity of open, half-open, and closed edges and sides—complete or partial.

The named interplay between geometry and topology of the metric spaces gets more and more complicated and less intuitive the more abstract and the more elaborated the metric spaces are. We do not expect any clarification, for example, metric spaces of functions before sub-spaces of the Euclidean spaces are understood.

Future research will concentrate on the question, what else can be said about plastic and non-plastic sub-spaces of the space (ℝ, d). Furthermore, we will explore the extension of a metric space A to larger sets in A_ex which contain A. In particular, the metric hull, i. e., the set

gives interesting perspectives in the context of the plasticity problem for the specification A_ex = ℝ. We conjecture that the metric hull is the smallest proper extension of the metric space, which is simply connected to A_ex and where the plasticity is dominated by the geometry. Therefore, the topology might be sub-ordinated. In the medium term, we hope for an insight into the question how geometry and topology interact in the plasticity of a metric space.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

OZ: Writing – review & editing, Writing – original draft, Project administration, Formal analysis, Conceptualization. DL: Validation, Writing – review & editing, Writing – original draft, Visualization, Methodology.