DOI: https://doi.org/10.62204/2336-498X-2025-2-9

FORMATION OF THE CONCEPT OF METRIC SPACE IN THE PROCESS OF TRAINING HIGHER EDUCATION STUDENTS IN MATHEMATICAL SPECIALTIES AT UNIVERSITIES

Natalia Shapovalova,
Ph.D. in Physico-mathematical Sciences, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
n.v.shapovalova@udu.edu.ua; ORCID: 0009-0000-7084-1460
Larisa Panchenko,
Ph.D. in Pedagogy, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
larpan97@gmail.com; ORCID: 0009-0001-8156-286X

Olga Mihulova,
Teacher of the highest category, senior teacher,
Gymnasium No. 258 of the Dniprovskyi district of Kyiv,
rukovoditel10b@gmail.com; ORCID: 0009-0008-9916-0560

Annotation. The article describes the evolution of the concept of metric space in mathematics, its development from the emergence of the basic metric concepts: length and distance. The axiomatic method of introducing metric spaces is considered. The feasibility of using the axiomatic method as a logical basis for constructing metric spaces is substantiated. The specifics of the application of the axiomatic method when introducing the concept of metric space in higher and secondary education institutions are described. Various concepts of distance are analyzed. The concepts of Kagan-Birkhoff, Euclid-Kolmogorov distance are described in detail. The main methodological aspects of these processes in higher and secondary education institutions are revealed. Conclusions are made regarding the need and prospects for research into metric spaces and their application in higher education institutions.

Keywords: axiomatic method, distance, length, metric space, geometry, educational process, learning. 

Problem statement. The evolution of the concept of metric space in mathematics can only be understood by tracing its development from the emergence of the basic metric concepts: length and distance.

The abstract-mathematical concept of «distance» is closely related to the abstract- mathematical concept of «metric space». These concepts are geometric in origin, and therefore their genesis is inseparable from the development of the idea of space and geometry. The idea of the shortest distance between two points arose among primitive people and, as some scientists believe, arose earlier than the concept of number. The concept of distance was formed in the process of solving practical problems related to measurements on the earth’s surface. In the process of repeatedly comparing the lengths of different lines connecting two different objects in space, comparing and analyzing the results, the concept of distance between two points was formed. Therefore, this concept is not genetically primitive.

The study of topics related to the theory of metric spaces, their use in various sections of modern mathematics, creates objective prerequisites for considering modern concepts of distance and metric space.

The study of such important concepts of analysis as limit and continuity in an arbitrary metric space makes them simpler and clearer. Considering a general metric space, we seem to forget for a while about the nature of its elements, and study only those of their properties that are related to the metric. As a result, with almost elementary logical markings we obtain facts that are applicable to many specific spaces at once [10, p.12].

High-quality study of these topics ensures the development of mathematical abilities and logical thinking of students. Solving problems on displacement plays a significant role in the formation of metric concepts. Such problems help students realize that geometric concepts and theorems reflect the spatial properties and relationships of real objects.

Analysis of recent research and publications. The basic properties of distance were already known in Assyria and Babylonia, Egypt. However, only Greek mathematicians gave geometry its characteristic abstract direction. One of the most prominent works of ancient Greek mathematics is the «Principles» of the famous Euclid of Alexandria (early 3rd century BC).

In this work, a geometric model of space is constructed for the first time. Here, the concept of «distance» is interpreted as a scalar quantity – the length (extension) of a segment connecting two given points.

The concept of three-dimensional Euclidean space was formed on the basis of people’s active perception of the properties of the surrounding space in the process of their practical activities – more precisely, the perception of individual real spatial objects. The development of navigation and astronomy led to the emergence of the concept of spherical space, in which the distance between two points is defined as the length of the smaller of the arcs of the great circle of the sphere passing through these points. Therefore, the creation of such metric spaces as Euclidean and spherical is the result of direct abstraction from the forms and relations of the real world.

However, further on, the concept of space in mathematics develops indirectly: already existing geometric abstractions are generalized.

In the first axiomatically constructed metric spaces, distance is not yet a primitive concept whose properties are directly given by a system of axioms.

The impetus for the generalization of the concept of distance was the creation by René Descartes and Pierre Fermat of the coordinate method, which allowed them to establish a connection between a set of points in three-dimensional space and a set of ordered triples of numbers.

Further development of the concept of multidimensional space occurs in the process of solving various practical problems, in particular those that are reduced to solving a system of equations with many variables. The geometry of n-dimensional space becomes a powerful tool both for describing the states of various systems in physics and chemistry, and for elucidating new properties of the objective world. It has opened wide avenues for algebra and analysis in the methodology of proofs.

Thus, the concepts of distance and metric space arose as an abstraction upon abstraction as a result of the comparison of abstract concepts already reduced to mathematics. Having lost their materialized concreteness, the abstract-mathematical concepts of distance and metric space acquired great generality. For example, Euclidean and non-Euclidean spaces (spherical, hyperbolic, elliptic) are just special cases of metric space, since their metric can be viewed as a specific interpretation of its axioms.

Issues related to the use of the concept of metric space and the study of the properties and applications of metric spaces in the study of mathematical analysis, geometry, and topology were addressed by such famous scientists as René Fréchet, who in 1906 first considered the idea of an abstract space with metric properties, Felix Hausdorff, who in 1914 introduced the term «metric space», D.Yu. Burago, Y.D. Burago, S.V. Ivanov, who are the authors of a book on metric geometry, N. Vasiliev, Y.A. Schreider, S. Banakh, Y.M. Berezansky, M.I. Zhaldak, G.O. Mikhalin and others.

The purpose of the article. The purpose of the article is to describe, using available sources, scientific and methodological literature, and my own experience, the evolution of the concept of metric space in mathematics, its development from the emergence of the basic metric concepts: length and distance. To reveal the essence and justify the feasibility of using the axiomatic method of constructing metric space. Describe the specifics of the application of the axiomatic method when introducing the concept of metric space in higher and secondary education institutions. Analyze different concepts of distance. Reveal the main methodological aspects of these processes in higher and secondary education institutions.

Presenting main material. At the current stage of development of mathematical science, the axiomatic definition of a metric space is given based on the mapping of the Cartesian (direct) product of the set M onto the set of real non-negative numbers.

Example. Consider a finite set of spectators in a movie theater and a set of chairs. If the theater is full of spectators and all the chairs are occupied, then we can say that the set of spectators is mapped onto the set of chairs.

Indeed, in this case, each spectator х ∈ А corresponds to a certain chair у ∈ В. We have the mapping: А→В. If some chairs in the cinema are empty, then from a mathematical point of view, the set A is mapped to the set B. In the first case, for each element y in the set B there is a preimage x in the set A. In the second case, not for each element y there is a preimage x.

Therefore, a mapping of set A into set B is a correspondence in which each element of set A corresponds to one element of set B.

f

Mappings are denoted by А ® В, or х→f(x); x ∈ A, f(x) ∈ B. The set of all images

of elements of a set A is denoted by f(A). If f(A))=B, then we say that there is a mapping of the set A onto the set B. If A=B, then we say that f is a mapping of the set A onto itself, or a transformation of the set A into itself.

If, in a mapping f of a set A onto a set B, every two different elements of the set A

correspond to different elements of the set B, then such a mapping is called one-to-one,

or invertible.                            f

Example. A mapping R ® E of the set R of real numbers onto the set E of points on a Euclidean line is one-to-one, such that each point M of the line corresponds to a unique real number хм ΠR. (The length of the unit segment is fixed).

Example. Taking two mutually perpendicular lines on the Euclidean plane and

introducing a Cartesian coordinate system, we define a mapping of a set of pairs of real numbers onto a set of points of this plane. Indeed, each pair of numbers (x; y) corresponds to a unique point M – the intersection of lines drawn through the points Mx and My parallel to the given coordinate axes Ox and Oy. In this case, we speak of the mapping of the Cartesian product of the set of real numbers onto the set of points of the Euclidean plane.

The Cartesian, or direct, product of a set A by a set B is called the set of all possible ordered pairs of elements of sets A and B.

Therefore, А×В = { (x; y) | x ΠА, у ΠВ}. The concept of the Cartesian product of

some sets A and B is very important in mathematics, since it is based on which a number of important mathematical concepts are introduced: algebraic operation, metric function, and mathematical space.

Example. Suppose we want to estimate the approximate distance in a straight line between two cities, depicted on a map by points. First, we measure this distance with a scale ruler and assign a certain positive number to the pair of points P1 and P2 on the map, which expresses the actual distance on the map. By multiplying the found number by the scale, we estimate the approximate distance we are looking for. So, we have here a mapping of pairs of points of a geographic map onto the set of real positive numbers, or a mapping of the Cartesian product K×K of the set of points of a geographic map onto the set of real positive numbers.

This mapping satisfies the distance properties:

• ρ (П1, П2)>0 ⇔ П1 ¹П2 і ρ (П1, П2)=0 ⇔ П1=П2 ;
• ρ (П1, П2) = ρ (П2, П1);
• ρ (П1, П2)< ρ (П1, П3) + ρ (П3, П2).

In this case, we say that we are dealing with a metric space. Thus, a metric space

constructively consists of two mathematical objects – some non-empty set M and a mapping ρ, which associates each pair of elements (x; y) from the set M with a real non- negative number ρ (x; y) such that conditions 1) – 3) are satisfied.

Definition 1. A metric space is an arbitrary nonempty set M such that for the Cartesian product M×M it is given by a mapping ρ onto the set of real nonnegative numbers, which associates each pair (x; y) ∈ M×M with the number ρ (x; y), and the following conditions are satisfied [6, p.7]:

• ∀ х, у ∈ М , ρ (х ; у)>0 ⇔ x≠ y і ρ (х ; у)=0 ⇔ х = у;
• ∀ х, у ∈ М , ρ (х ; у)= ρ ( у; х) (axiom of symmetry);
• ∀ х, у, z ∈ М , ρ (х ; у) ≤ ρ (х ; z)+ ρ (z; y) (triangle axiom).

The mapping ρ is called a metric function, or metric, and the numerical value of the metric function for given x and y is the distance from point x to point y.

Properties 1) – 3) are called the axioms of metric space (Fréchet axioms):

• the distance between two distinct points is a positive real number function if the points are distinct. The distance of a point to itself is zero;
• the distance from the first point to the second is equal to the distance from the second point to the first;
• if, in addition to the two given points, we take some third point different from the given ones, then the distance between the two given points does not exceed the sum of their distances to this point.

A metric space M with a given distance ρ on it is a mathematical structure, which is denoted by (M; ρ).

Definition 2. A model of a metric space is a set of specific objects and relations between them that have the properties listed in axioms 1 – 3.

Example. On the shore of a lake (Fig. 1) there are three piers A, B, and C, between which a steamboat travels at a constant average speed. Let us call these piers points, and for the distance from one point X to the second Y we take the time tx y, during which the

steamboat covers the distance from pier X to pier Y. It is easy to verify that the distance

introduced in this way has properties 1-3:

1. ρ (Х ; У)= txy > 0 ⇔ X≠ Y  і  ρ (Х ; Y)= tx y =0  ⇔ Х = Y;
2. tx y = ty x ;
3. Since the steamer moves at a constant speed, the time of its movement is

proportional to the distance.

Here always

S x y< S x z +S z y , and therefore t x y < t x z + t z y

(X, Y, Z – any of the piers А, В, С)

So, here each pair of points of the set {A, B, C} is mapped to some subset of the set of real non-negative numbers such that the properties of the metric 1 – 3 are fulfilled. We have a model of a metric space, where time is taken as the distance.

We can also give the following definition of a metric space:

Definition 3. An arbitrary set X of elements is called a metric space if for any two elements x and y of this set there is associated a non-negative function ρ (х; у), which is called a distance or metric, satisfying the following conditions:

• ρ (х; у)=0 ⇔ х = у;
• ρ (х; у)= ρ ( у; х);
• ρ (х; у) ≤ ρ (х; z)+ ρ (z; y) .

Let us consider examples of metric spaces.

1. The space R n is a Euclidean n-dimensional space [6, 9-10]. The metric in this space is introduced by the formula

As a special case, we consider the space R1, the metric in which is introduced by the formula

ρ (х; у)=|x-y|.

This metric is called the standard metric.

1. The space С[a,b] of continuous functions on the interval [a, b] [5, 19]. In this space, the metric is introduced as follows:

On the set of continuous functions, we can introduce the following metric:

Let us give some information from the theory of metric spaces.

Theorem. Each sequence of points in a metric space (M, ρ) can have only one boundary.

Definition 4. A sequence of points in a metric space that has a boundary is called

convergent [10, p. 24].

Definition 5. A sequence (хn) is called fundamental if for every ε > 0 there exists a number n₀ (ε):

ρ ( хn , хm)< ε      ∀ n , m >n0 (ε).

Theorem. Every sequence that is fundamental is a bounded sequence.

Theorem. If a sequence (хn) is convergent in a metric space M, then it is fundamental in this space.

Definition 6. A metric space (M, ρ) is called complete if every fundamental sequence in (M, ρ) has a limit [10, p.59].

The concept of distance. Euclidean metric space. The concept of distance Kagan – Birkhoff.

The concept of distance in its classical form is interpreted as a real non-negative number. This interpretation of distance was proposed by the Soviet mathematician

1. Kagan (1869-1953) in his work «A System of Assumptions Defining Euclidean Geometry» (1902). Kagan’s axiom system of Euclidean geometry is based on the concept of distance as an invariant of the group of axioms of displacements, and distance is interpreted as a real non-negative number.

His idea was developed when constructing a geometry course for secondary education institutions. This idea was most fully embodied in the work «A System of Axioms of Planimetry Based on the Use of a Scale Ruler and Protractor» (1932), by the outstanding American mathematician and educator D. Birkhoff (1884–1944). The Birkhoff system is widely used in American secondary education institutions.

According to Kagan-Birkhoff, the concept of distance can be axiomatically defined as follows:

1. For each pair of points A and B, a distance is defined, denoted by ρ (А; В);
2. The distance ρ ρ (А; В) is a non-negative real number ρ (А; В) ∈ R+ ;
   1. ∀ (А; В) [ρ (А; В)=0 ⇔ А=В];
   2. ∀ (А; В) [ρ (А; В)= ρ (В; А)];
   3. ∀ (А; В; С) [ρ (А; В)+ ρ (В; С) ≥ ρ (А; С)].

The Euclidean-Kolmogorov concept of distance. The concept of distance, originating from Euclid and developed in the works of Kolmogorov, is a concept based on a clear distinction between a geometric figure as a carrier of a quantity, the quantity itself, and its numerical value – a non-negative real number.

In this concept, distance is considered as a non-negative scalar quantity. A system of homogeneous scalar quantities is defined as a mathematical structure (S, +, *, >), consisting of a set S of quantities of this system and the binary operations of addition, multiplication by a real number and the relation «>» specified on it, and:

• Among the quantities S there is a quantity Ō (zero scalar) such that always

x Ō = Ō, where x is any real number, i.e. ∃! (Ō∈ S) ∀ (x ∈R) (x Ō = Ō)$.

• For any positive scalar quantity ē ∈ S the mapping f: х → ā = х ē one-to-one maps the set of real numbers R onto the set S.
• ∀ х,у ∈ R ∀ ē ÎS х ē + у·ē = (х+у) ē ;
• ∀ х,у ∈ R ∀ ē ÎS х ( у·ē )= (х·у) ē ; 5,) ∀ х,у ∈ R ∀ ē ∈S x/y=xe/ye.

Introducing the concept of «quantity» into the mathematics course of secondary education institutions makes it possible to approach the issue of measuring lengths, areas, and volumes from an objective perspective, considering quantity as a geometric property that can be quantitatively characterized by a real number.

In the Euclid-Kolmogorov concept, the concept of distance is axiomatically defined as follows:

• For each pair of points A and B, a distance is defined, which is denoted by |AB|.

• The distance |AB| is a non-negative scalar
• ∀ (А,В) |АВ|=|ВА|;
• ∀ (А,В,С) |АВ|+|ВС| ³|АС|.

Distance exists independently of the measurement process.

When measuring the length of a segment or the distance between its ends, we deal with three sets:

• the set of geometric shapes – segments [XY];
• the set of quantities – lengths of segments [XY];
• the set of numerical values of the lengths of segments – real non-negative numbers ρ |ХY|.

The interpretation of distance as a quantity is the basis of the axiomatics of the geometry course in secondary education institutions, proposed by Academician A.M. Kolmogorov.

The main undefined concepts here are:

− “point”, “distance”, “line”, – in Euclidean planimetry;

− “point”, “distance”, “line”, “plane”, – in Euclidean stereometry. Kolmogorov’s axiom system consists of 14 axioms and is divided into 5 groups Group 1 – membership axioms;

Group 2 – distance axioms; Group 3 – order axioms;

Group 4 – plane motion axioms; Group 5 – parallel axioms.

The group of distance axioms consists of the following axioms:

Axiom 1. For any two points A and B, there is a non-negative quantity called the distance from A to B. The distance AB is zero if and only if the points A and B coincide. Axiom 2. The distance from point A to point B is equal to the distance from point B to point A.

Axiom 3. For any three points A, B, C, the distance from point A to C is no greater than the sum of the distances from point A to point B and from point B to point C.

If in these axioms we replace the distance as a quantity with its numerical value, we obtain the axioms of metric space.

It follows that the two-dimensional and three-dimensional Euclidean spaces, built on the basis of Kolmogorov’s axioms, are metric.

The distance between points and figures in Euclidean metric space. People began to deal with problems of finding distances in ancient times. By improving the methods of solving these problems and the techniques used, people achieved great mastery in this matter: they learned to measure cosmic distances and distances between atomic particles. Analysis of solutions to problems for finding distances, construction of various measurement models led to the emergence of the concept of metric space.

Let’s consider problems for finding distances between figures using a geometric method. The essence of this method is that we connect the points, the distance between which needs to be found, with a straight-line segment and take the length of this segment as the desired distance, to which we can compare a certain (with the selected unit of measurement) numerical characteristic – the numerical value of the distance.

How to find the distance from a point to a figure? From one figure to another? These are important questions. Their clarification is necessary for descriptive geometry, drawing, cartography and other applied sciences. The correct solution to these questions can be found based on the fact that geometric figures are point sets. Therefore, you can use the concept of distance between two sets

Definition. The distance from point A to figure F is called the smallest (if it exists) of the distances from point A to all points of figure F. Such a smallest distance does not always exist. For example, there is no distance from point A to the interval BC if points A, B, C lie on the same straight line, or the orthogonal projection of point A does not belong to the segment BC. (Fig. 2)

Algorithm for finding the distance from a given point to a given convex polygon by construction:

• find the sides of the polygon (no more than two), such that the lines defined by them divide the plane into two half-planes, and the given polygon and the given point lie in different half-planes;
• from the given point, draw a perpendicular to the selected sides;
• if the base of one of the constructed perpendiculars belongs to a side of the polygon, then the length of the segment of this perpendicular is taken as the desired distance;
• if the base of none of the perpendiculars belongs to the sides of the polygon, then the desired distance is taken as the segment connecting the given point with the vertex of the polygon.

Based on this, problems are solved to find sets of points that have certain (metric) properties. Such problems are used in various branches of modern mathematics. For example, in geometry, figures are defined as sets of points that have certain properties.

When solving problems to construct a figure as a set of points with certain metric properties, the role of the drawing also increases. The system of problems should ensure their gradual complication.

Definition. The distance from the figure F₁ to the figure F₂ is called the smallest (if it exists) of the distances from all points of the figure F₁ to all points of the figure F₂.

This distance also does not always exist. For example, there is no distance between two intervals that lie on the same line, between the graph of an exponential function and the abscissa axis. (Fig. 3)

Since we consider a geometric figure as a set of points, it makes sense to pose the problem of finding distances if the intersection of the figures is an empty set; the distance between figures whose intersection is a non-empty set is considered to be zero.

Only by considering the internal metric can one clarify the question of the distance between two points on some surfaces of three-dimensional Euclidean space. For this, it is advisable to use the methods of synthetic geometry, which are closer to the practice of teaching in secondary education institutions.

Finding the shortest distances between two points on polyhedral surfaces in secondary education institutions is reduced to constructing scans of these surfaces.

Problem. The hall has dimensions of 12×12×30 m. On one of the smaller walls in the middle, at a distance of 1 m from the floor, there is a fly sitting. On the opposite wall is a spider. What is the shortest path the spider must take to catch the fly?

Most often, this problem is solved as shown in Figure 4 and the answer is:

|AB|=42 m. But this solution is incorrect, because there are many different options for constructing the sweep, in particular, the one shown in Figure 5. In this case, the spider’s path will be represented by the segment AB:

Let us consider some questions of the internal metric of cylindrical, conical and spherical surfaces. These surfaces of revolution are considered in the course of geometry in secondary education institutions. They are an illustration of the possibilities of different definitions of the distance between two points that is accessible to students.

Helical lines on the surface of a cylinder and cone are lines that intersect its generatrix at angles whose magnitudes are equal to each other but different from 90⁰. The shape of a helical line is the thread on drills, nuts, and bolts. Beans, peas, hops, and grapes wrap around a cylindrical support along a helical line, which is the shortest.

The most important statements.

1. The shortest geodesics on the surface of a cylinder are the generators, helices, and the smaller of the arcs of parallels connecting two given points.
2. Parallels and helices are not the shortest on the surface of a cone. In addition to the generators, there are various shortest geodesics that can be found by constructive
3. The shortest on the surface of a sphere are the smaller of the arcs of great circles passing through two given points on the sphere.

Compressive mappings and their applications. One of the most important results in the theory of metric spaces is Banach’s theorem on the properties of a contracting map in a complete metric space. This theorem characterizes the completeness of a metric space and has extremely wide applications. Using it, one can prove the existence and uniqueness of solutions to various algebraic and transcendental equations, systems of algebraic equations, differential and integral equations, etc. Moreover, it provides a method for finding this approximate solution. This is the so-called method of successive approximations.

Conclusions. People began to deal with problems of finding distances in ancient times. Improving the methods of solving these problems and the technique used in this, they achieved great skill in this matter, learned to measure the distances to stars and the distances between atomic particles. Analysis of solutions to problems of finding distances, construction of various measurement models led to the emergence of the concept of metric space.

The concepts of distance and metric space occupy a special place in the entire system of scientific knowledge. The most important applications to which the theory of metric spaces owes its emergence and development are related to geometry and topology, analysis and function theory. It is worth noting the application of the theory of metric spaces to the proof of Peano’s theorem on the existence of a solution to the Cauchy problem, the Stone-Weierstrass theorem and the corollary on the approximation of a function by polynomials, and the construction of the theory of integrable functions by completing the incomplete space of step functions [10, section 7, p. 346].

Many fundamental facts of geometry, topology, and analysis are based on the properties of real numbers, which use the concept of distance. For example, one of the most important operations of mathematical analysis, the operation of limit transition, is based on the concept of distance between points on a number line (plane, three- dimensional space).

It is especially worth noting the connection between metric and topological spaces, since every metric space is topological.

Multidimensional metric spaces have found wide application in various branches of physics. Thus, the starting point for the construction of classical statistical physics is the idea of phase space – the space of all generalized coordinates and all generalized momenta of the considered physical system.

The multidimensional phase space method is used in mechanics, thermodynamics, and physical chemistry.

The concept of metric space helps students to understand the essence of the coordinate method more deeply; apply the acquired knowledge to solving problems using the coordinate method, solving equations and inequalities containing modules. Properly selected problems will help students understand the practical need for studying metric spaces.

The topic of metric spaces deserves close attention from lecturers, teachers, and students due to its deep theoretical content, which contributes to the development of mental abilities.

The study of metric spaces in mathematics courses in higher education institutions is becoming especially relevant due to their widespread application not only in mathematics, but also in physics, economics, biology, medicine, and other fields.

It is advisable to give practical meaning to problems for finding distances between figures, which are solved by the method of displacements. Then the concept of distance is specified through objects of surrounding reality, which creates a positive emotional attitude towards mathematics in students.

Improving the assimilation of topics will be facilitated primarily by:

• inclusion of various interesting tasks and tasks that attract the attention of students with elements of novelty and Emotional upliftment increases the cognitive abilities of students, stimulates the activity of mental actions, and increases the productivity of memory;

• optimal selection of content, methods, organizational forms and means of learning, taking into account age characteristics, time allocated to studying the topic, level of knowledge and skills of education seekers.

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