What is topology

Introduction

1. Main stages in the development of topology

2. General characteristics of the topology

3. General topology

4. Topological space

5. Important issues and results

Conclusion

Introduction

Topology is a relatively young mathematical science. For about a hundred years of its existence, it has achieved results that are important for many branches of mathematics. Therefore, penetration into the “world of topology” is somewhat difficult for a beginner, since it requires knowledge of many facts of geometry, algebra, analysis and other branches of mathematics, as well as the ability to reason.

Topology influences many branches of mathematics. It studies, in particular, such properties of arbitrary geometric images that are preserved under transformations that occur without breaks and gluing, or, as mathematicians say, under one-to-one and mutually continuous transformations. Such transformations are called topological. Two geometric images in a topology are considered "the same" if one of them can be translated into the other by a topological transformation. For example, a circle and a square on a plane can be transformed into each other by a topological transformation - these are topologically equivalent figures. At the same time, a circle and an annular area, obtained from a circle by “ejecting” a concentric circle of a smaller radius, are different from the point of view of topology.

Topology is divided into two sections - general or set-theoretic topology and algebraic topology. The division is largely conditional. One of the main tasks of general topology is the analysis of the mathematical concept of continuity in its most general form. For this, the concept of a topological space was introduced. Topology has developed a highly sophisticated algebraic and analytic technique that extends far beyond its original scope. This includes, in particular, the so-called homological algebra, which is also a working tool in the theory of partial differential equations, in the theory of functions of many complex variables, etc. One of the branches of general topology is the dimension theory. What does it mean that some space is two-dimensional, three-dimensional, or, in general, n-dimensional? Dimension is one of the fundamental characteristics of a topological space. Its definition in the general case turns out to be very difficult. V. Kuzminov constructed a number of examples showing the paradoxical behavior of dimension in certain situations. I. Shvedov studied the problem of the axiomatic definition of dimensions, and he refuted, in particular, some well-known conjectures related to this problem. Another branch of topology is called Hodge theory. This theory combines ideas related to the theory of partial differential equations, Riemannian geometry and topology. V. Kuzminov, I. Shvedov, and V. Goldstein in a series of papers constructed a certain generalization of the Hodge theory applicable to the study of manifolds with singularities and manifolds satisfying reduced (in comparison with the usual Hodge theory) smoothness requirements. The difference of this generalized Hodge theory, from the point of view of differential equations, is that this theory is essentially non-linear.

1. Main stages in the development of topology

2. General characteristics of the topology

One of the most unexpected phenomena in the development of mathematics of the XX century. was the dizzying rise of the science known as topology.

Topology (from the Greek τόπος - place and λόγος - word, doctrine) is a section of geometry that studies the phenomenon of continuity in its most general form, in particular the properties of space that remain unchanged under continuous deformations, for example, connectivity, orientability.

In order to clarify what topology is, it is sometimes said that it is "geometry on a rubber surface." This obscure and vague description, however, allows you to capture the essence of the subject. Topology studies those properties of geometric objects that are preserved under continuous transformations. Continuous transformations are characterized by the fact that the points located "close to each other" before the transformation remain so after the transformation is completed. During topological transformations, it is allowed to stretch and bend, but it is not allowed to break and tear. (However, with one caveat: when it comes to transformations, we are not interested in what happens in the process of these transformations, only the initial position and the final result are important. Therefore, it is allowed, say, cuts along some lines, which are then glued along the same lines (for example, if the shoelace is knotted and its ends are connected, you can cut it somewhere, untie the knot and reconnect it at the cut point).

Topology can be divided into three areas:

1) combinatorial topology, which studies geometric shapes by dividing them into simple figures that adjoin each other in a regular way;

2) algebraic topology, which deals with the study of algebraic structures associated with topological spaces, with an emphasis on group theory;

3) set-theoretic topology, which studies sets as clusters of points (as opposed to combinatorial methods, which represent an object as a union of simpler objects) and describes sets in terms of such topological properties as openness, closedness, connectedness, etc. Of course, such a division of topology into regions is somewhat arbitrary; many topologists prefer to single out other sections in it.

What kind of properties are topological? Clearly, not those that are studied in ordinary Euclidean geometry. Straightness is not a topological property, because a straight line can be bent and become wavy. A triangle is also not a topological property, because a triangle can be continuously deformed into a circle.

So, in topology, a triangle and a circle are one and the same. The lengths of the segments, the magnitude of the angles, the areas - all these concepts change with continuous transformations, and they should be forgotten. Very few familiar concepts of geometry are suitable for topology, so we have to look for new ones. This makes topology difficult for beginners until they get the gist of it.

An example of the topological property of an object is the presence of a hole in a donut (moreover, a rather subtle side of this matter is the fact that the hole is not part of the donut). Whatever continuous deformation the donut undergoes, the hole will remain. There is a catchphrase that a topologist (a mathematician who deals with topology) is a person who does not distinguish a donut from a teacup. This means that the most general (topological) properties of a donut and a cup are the same (they are solid and have one hole).

Another topological property is the presence of an edge. The surface of a sphere has no edge, but an empty hemisphere does, and no continuous transformation can change that.

The main objects of study in topology are called topological spaces. Intuitively, they can be thought of as geometric shapes. Mathematically, these are sets (sometimes subsets of the Euclidean space), endowed with an additional structure called topology, which allows us to formalize the concept of continuity. The surface of a sphere, a donut (more correctly, a torus), or a double torus are examples of topological spaces.

Two topological spaces are topologically equivalent if it is possible to go continuously from one of them to the other and go back in a continuous way.

We have to introduce the requirement of continuity, both direct mapping and its inverse, for the following reason. Take two pieces of clay and stick them together. Such a transformation is continuous, since points close to each other will remain so.

However, with the reverse transformation, one piece splits into two, and, consequently, close points on opposite sides of the dividing line will be far from each other, i.e. the inverse transformation will not be continuous. Such transformations do not suit us.

Geometric figures that pass one into another under topological transformations are called homeomorphic. The circle and the boundary of a square are homeomorphic, since they can be transformed into each other by a topological transformation (i.e., by bending and stretching without breaking or gluing, for example, stretching the boundary of a square by the circumscribed circle). The sphere and the surface of a cube are also homeomorphic. To prove that figures are homeomorphic, it suffices to indicate the corresponding transformation, but the fact that we cannot find a transformation for some figures does not prove that these figures are not homeomorphic. Topological properties help here.

TOPOLOGY
branch of mathematics concerned with the study of the properties of figures (or spaces) that are preserved under continuous deformations, such as, for example, tension, compression, or bending. Continuous deformation is a deformation of a figure, in which there are no breaks (i.e., violation of the integrity of the figure) or gluing (i.e., identification of its points). Such geometric properties are related to the position, and not to the shape or size of the figure. Unlike Euclidean and Riemannian geometries, Lobachevsky geometry and other geometries dealing with the measurement of lengths and angles, topology has a non-metric and qualitative character. Previously, it was called "situs analysis" (position analysis), as well as "point set theory". In the popular science literature, topology is often referred to as "rubber sheet geometry" because it can be visualized as the geometry of shapes drawn on perfectly elastic rubber sheets that are stretched, compressed, or bent. Topology is one of the newest branches of mathematics.
Story. In 1640 the French mathematician R. Descartes (1596-1650) found an invariant relation between the number of vertices, edges and faces of simple polyhedra. Descartes expressed this relation by the formula V - E + F = 2, where V is the number of vertices, E is the number of edges and F is the number of faces. In 1752 the Swiss mathematician L. Euler (1707-1783) gave a rigorous proof of this formula. Another contribution of Euler to the development of topology is the solution of the famous Königsberg bridge problem. It was about an island on the Pregel River in Koenigsberg (in the place where the river divides into two branches - Old and New Pregel) and seven bridges connecting the island with the banks. The challenge was to find out if it was possible to bypass all seven bridges in a continuous route, visiting each one only once and returning to the starting point. Euler replaced land areas with points and bridges with lines. Euler called the resulting configuration a graph, points - its vertices, and lines - edges. He divided the vertices into even and odd, depending on whether an even or odd number of edges emerge from the vertex. Euler showed that all edges of a graph can be traversed exactly once along a continuous closed path only if the graph contains only even vertices. Since the graph in the Königsberg bridge problem contains only odd vertices, it is impossible to go around the bridges along a continuous route, visiting each exactly once and returning to the beginning of the route. Euler's proposed solution to the problem of Königsberg bridges depends only on the relative position of the bridges. It marked the formal beginning of topology as a branch of mathematics. K. Gauss (1777-1855) created the theory of knots, which was later studied by I. Listing (1808-1882), P. Tate (1831-1901) and J. Alexander. In 1840 A. Möbius (1790-1868) formulated the so-called four-color problem, which was later investigated by O. de Morgan (1806-1871) and A. Cayley (1821-1895). The first systematic work on topology was Listing's Preliminary Studies on Topology (1874). The founders of modern topology are G. Kantor (1845-1918), A. Poincaré (1854-1912) and L. Brouwer (1881-1966).
Topology sections. Topology can be divided into three areas: 1) combinatorial topology, which studies geometric shapes by breaking them down into simple shapes that adjoin each other in a regular way; 2) algebraic topology, which deals with the study of algebraic structures associated with topological spaces, with an emphasis on group theory; 3) set-theoretic topology, which studies sets as clusters of points (as opposed to combinatorial methods, which represent an object as a union of simpler objects) and describes sets in terms of such topological properties as openness, closedness, connectedness, etc. Of course, such a division of topology into regions is somewhat arbitrary; many topologists prefer to single out other sections in it.
Some basic concepts. The topological space consists of a set of points S and a set S of subsets of the set S satisfying the following axioms: (1) the entire set S and the empty set belong to the set S; (2) the union of any collection of sets from S is a set from S; (3) the intersection of any finite number of sets from S is a set from S. The sets included in the set S are called open sets, and this set itself is called a topology in S.
See SET THEORY. A topological transformation, or homeomorphism, of one geometric figure S to another, S", is a mapping (p (r) p") of points p from S to points p" of S", satisfying the following conditions: 1) the correspondence between the points established by it from S and S" is one-to-one, i.e. each point p from S corresponds to only one point p" from S" and only one point p is mapped to each point p"; 2) the mapping is mutually continuous (continuous in both directions), i.e. if two points p, q from S are given and the point p moves so that the distance between it and the point q tends to zero, then the distance between the corresponding points p", q" from S" also tends to zero, and vice versa. Geometric figures, The circle and the boundary of a square are homeomorphic, since they can be transformed into each other by a topological transformation (i.e., by bending and stretching without breaking or gluing, for example, by stretching the boundary of a square by the circumscribed circle The sphere and the surface of the cube are also homeomorphic. To prove the homeomorphism of figures, it is enough to indicate the corresponding transformation, but the fact that we cannot find a transformation for some figures does not prove that these figures are not homeomorphic. Topological properties help here.

Rice. 1. THE SURFACE OF THE CUBE AND THE SPHERE are homeomorphic, i.e. can be translated into each other by a topological transformation, but neither the surface of the cube nor the sphere are homeomorphic to the torus (surfaces of the "donut").

A topological property (or topological invariant) of geometric figures is a property that, together with a given figure, also has any figure into which it passes during a topological transformation. Any open connected set containing at least one point is called a region. A domain in which any closed simple (that is, homeomorphic to a circle) curve can be contracted to a point while remaining in this domain all the time is called simply connected, and the corresponding property of the domain is simply connected. If, however, some closed simple curve of this region cannot be contracted to a point, remaining all the time in this region, then the region is called multiply connected, and the corresponding property of the region is multiply connected. Imagine two circular regions, or disks, one with no holes and one with holes. The first region is simply connected, the second is multiply connected. Simply connected and multiply connected are topological properties. A region with a hole cannot pass under a homeomorphism to a region without holes. It is interesting to note that if in a multiply connected disk a cut is drawn from each of the holes to the edge of the disk, then it becomes simply connected. The maximum number of closed simple non-intersecting curves along which a closed surface can be cut without dividing it into separate parts is called the genus of the surface. The genus is a topological invariant of a surface. It can be proved that the genus of a sphere is equal to zero, the genus of a torus (the "donut" surface) is one, the genus of a pretzel (a torus with two holes) is two, and the genus of a surface with p holes is equal to p. This implies that neither the surface of the cube nor the sphere is homeomorphic to the torus. Among the topological invariants of a surface, one can also note the number of sides and the number of edges. A disk has 2 sides, 1 edge, and genus 0. A torus has 2 sides, no edges, and its genus is 1. The concepts introduced above make it possible to refine the definition of topology: topology is a branch of mathematics that studies properties that are preserved under homeomorphisms.
Important problems and results. Jordan closed curve theorem. If a simple closed curve is drawn on the surface, is there any property of the curve that is preserved when the surface is deformed? The existence of such a property follows from the following theorem: a simple closed curve on a plane divides the plane into two regions, internal and external. This seemingly trivial theorem is obvious for curves of a simple form, for example, for a circle; however, the situation is different for complex closed broken lines. The theorem was first formulated and proved by K. Jordan (1838-1922); however, Jordan's proof turned out to be flawed. A satisfactory proof was proposed by O. Veblen (1880-1960) in 1905.
Brouwer's fixed point theorem. Let D be a closed region consisting of a circle and its interior. Brouwer's theorem states that for any continuous transformation that takes every point of the domain D to a point in the same domain, there is some point that remains unchanged under this transformation. (The transformation is not assumed to be one-to-one.) Brouwer's fixed-point theorem is of particular interest because it seems to be the most frequently used topological theorem in other branches of mathematics.
The problem of four colors. The problem is this: can any map be colored in four colors so that any two countries that share a common border are colored in different colors? The four-color problem is topological, since neither the shape of the countries nor the configuration of the borders matter. The conjecture that four colors are sufficient for the corresponding coloring of any card was first stated in 1852. Experience has shown that four colors are indeed sufficient, but a rigorous mathematical proof could not be obtained for more than a hundred years. And only in 1976, K. Appel and V. Haken from the University of Illinois, having spent more than 1000 hours of computer time, achieved success.
Unilateral surfaces. The simplest one-sided surface is the Möbius strip, named after A. Möbius, who discovered its extraordinary topological properties in 1858. Let ABCD (Fig. 2a) be a rectangular strip of paper. If you glue point A with point B, and point C with point D (Fig. 2b), you get a ring with an inner surface, an outer surface and two edges. One side of the ring (Fig. 2b) can be painted. The painted surface will be bounded by the edges of the ring. The beetle can "circle the world" around the ring, staying either on a painted or unpainted surface. But if the strip is twisted half a turn before gluing the ends and glue point A with point C, and B with D, then a Mobius strip will be obtained (Fig. 2, c). This shape has only one surface and one edge. Any attempt to color only one side of the Möbius strip is doomed to failure, since the Möbius strip has only one side. A beetle crawling along the middle of a Möbius strip (without crossing the edges) will return to its starting point in the "upside down" position. When cutting the Möbius strip along the midline, it does not split into two parts.

Nodes. A knot can be thought of as a tangled piece of thin rope with connected ends, located in space. The simplest example is to make a loop out of a piece of rope, pass one of its ends through the loop and connect the ends. As a result, we get a closed curve that remains topologically the same, no matter how it is stretched or twisted, without tearing or gluing separate points. The problem of classifying knots according to the system of topological invariants has not yet been solved.
LITERATURE
Hu Si-chiang. Homotopy theory. M., 1964 Kuratovsky A. Topology, vols. 1-2. M., 1966, 1969 Spanier E. Algebraic topology. M., 1971 Aleksandrov P.S. Introduction to set theory and general topology. M., 1977 Kelly J. General topology. M., 1981

Collier Encyclopedia. - Open Society. 2000 .

Synonyms:

See what "TOPOLOGY" is in other dictionaries:

Topology… Spelling Dictionary

topology- Physical or logical distribution of network nodes. The physical topology defines the physical links (channels) between nodes. The logical topology describes the possible connections between network nodes. In local networks, the most common are three ... ...

In a broad sense, the area of ​​mathematics that studies topological. properties diff. math. and physical objects. Intuitively, to the topological include qualitative, stable properties that do not change with deformations. Mat. formalization of the idea of ​​a topological properties ... ... Physical Encyclopedia

Science, the study of localities. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. topology (gr. topos place, area + ... ology) a branch of mathematics that studies the most general properties of geometric shapes (properties, not ... ... Dictionary of foreign words of the Russian language

TOPOLOGY, a branch of mathematics that studies the properties of geometric shapes that remain unchanged under any deformation, squeezing, stretching, twisting (but without gaps and gluing). A cup with a handle is topologically equivalent to a bagel; cube, ... ... Scientific and technical encyclopedic dictionary

- (from the Greek topos place and ... logic) a branch of mathematics that studies the topological properties of figures, that is, properties that do not change under any deformations produced without gaps and gluing (more precisely, with one-to-one and continuous ... ... Big Encyclopedic Dictionary

TOPOLOGY, topologies, pl. no, female (from the Greek. topos place and logos teaching) (mat.). A part of geometry that studies the qualitative properties of figures (i.e., independent of such concepts as length, angles, straightness, etc.). Dictionary… … Explanatory Dictionary of Ushakov

Exist., Number of synonyms: 1 math (29) ASIS Synonym Dictionary. V.N. Trishin. 2013 ... Synonym dictionary

Topology is a branch of mathematics that studies the properties of geometric figures that do not change under deformations that occur without discontinuities. Dictionary of business terms. Akademik.ru. 2001 ... Glossary of business terms

IC topology- — [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English Russian Dictionary of Electrical Engineering and Power Engineering, Moscow, 1999] Electrical engineering topics, basic concepts EN integrated circuit layout ... Technical Translator's Handbook

Term network topology refers to the way computers are connected to a network. You may also hear other names - network structure or network configuration (This is the same). In addition, the concept of topology includes many rules that determine the location of computers, cable laying methods, methods for placing connecting equipment, and much more. To date, several basic topologies have been formed and settled. Of these, it can be noted tire”, “ring" and " star”.

Bus topology

Topology tire (or, as it is often called common bus or highway ) assumes the use of one cable to which all workstations are connected. The common cable is used by all stations in turn. All messages sent by individual workstations are received and listened to by all other computers connected to the network. From this stream, each workstation selects messages addressed only to it.

Advantages of bus topology:

• ease of setup;
• relative ease of installation and low cost if all workstations are located nearby;
• the failure of one or more workstations does not affect the operation of the entire network.

Disadvantages of bus topology:

• bus failures anywhere (cable break, network connector failure) lead to network inoperability;
• difficulty in troubleshooting;
• low performance - at any given time, only one computer can transfer data to the network, with an increase in the number of workstations, network performance drops;
• poor scalability - to add new workstations, it is necessary to replace sections of the existing bus.

It was according to the “bus” topology that local networks were built on coaxial cable. In this case, segments of a coaxial cable connected by T-connectors acted as a bus. The bus was laid through all the premises and approached each computer. The side output of the T-connector was inserted into the connector on the network card. Here's what it looked like: Now such networks are hopelessly outdated and everywhere replaced by a twisted-pair “star”, however, equipment for coaxial cable can still be seen in some enterprises.

Topology "ring"

Ring - This is a local network topology in which workstations are connected in series to each other, forming a closed ring. Data is transferred from one workstation to another in one direction (in a circle). Each PC acts as a repeater, relaying messages to the next PC, i.e. data is transferred from one computer to another as if by relay. If a computer receives data intended for another computer, it transmits them further along the ring, otherwise they are not transmitted further.

Advantages of ring topology:

• ease of installation;
• almost complete absence of additional equipment;
• the possibility of stable operation without a significant drop in the data transfer rate during intensive network loading.

However, the “ring” also has significant drawbacks:

• each workstation must actively participate in the transfer of information; in the event of failure of at least one of them or a cable break, the operation of the entire network stops;
• connecting a new workstation requires a short network shutdown, since the ring must be open during the installation of a new PC;
• complexity of configuration and customization;
• difficulty in troubleshooting.

Ring network topology is rarely used. It has found its main application in fiber optic networks token ring standard.

Star topology

Star is a local network topology where each workstation is connected to a central device (switch or router). The central device controls the movement of packets in the network. Each computer is connected to the switch via a network card with a separate cable. If necessary, you can combine several networks with a star topology together - as a result, you will receive a network configuration with treelike topology. Tree topology is common in large companies. We will not consider it in detail in this article.

Topology "star" today has become the main one in the construction of local networks. This happened due to its many advantages:

• the failure of one workstation or damage to its cable does not affect the operation of the entire network as a whole;
• excellent scalability: to connect a new workstation, it is enough to lay a separate cable from the switch;
• easy troubleshooting and network interruptions;
• high performance;
• ease of setup and administration;
• additional equipment is easily integrated into the network.

However, like any topology, the “star” is not without its drawbacks:

• the failure of the central switch will result in the inoperability of the entire network;
• additional costs for network equipment - a device to which all network computers (switch) will be connected;
• the number of workstations is limited by the number of ports in the central switch.

Star - the most common topology for wired and wireless networks. An example of a star topology is a twisted-pair cable network with a switch as the central unit. These networks are found in most organizations.

Under topology(layout, configuration, structure) of a computer network is usually understood as the physical arrangement of network computers one relative to one and the way they are connected by communication lines. It is important to note that the concept of topology refers, first of all, to local networks, in which the structure of connections can be easily traced. In WANs, the link structure is usually hidden from users and is not very important, because each communication session can follow its own path.
The topology determines the requirements for the equipment, the type of cable used, the possible and most convenient methods of managing the exchange, the reliability of operation, and the possibility of expanding the network.

There are three main network topologies:

1. Bus network topology(bus), in which all computers are connected in parallel to one communication line and information from each computer is simultaneously transmitted to all other computers (Fig. 1);

2. Network topology star(star), in which other peripheral computers are connected to one central computer, and each of them uses its own separate communication line (Fig. 2);

3. Network topology ring(ring), in which each computer always transmits information to only one computer, the next one in the chain, and receives information only from the previous computer in the chain, and this chain is closed in a “ring” (Fig. 3).

Rice. 1. Network topology "bus"

Rice. 2. Network topology "star"

Rice. 3. Network topology "ring"

In practice, combinations of the basic topology are often used, but most networks are focused on these three. Let us now briefly consider the features of the listed network topology.

Bus topology(or, as it is also called, “common bus”) by its very structure allows the identity of the network equipment of computers, as well as the equality of all subscribers. With such a connection, computers can only transmit in turn, because there is only one communication line. Otherwise, the transmitted information will be distorted as a result of overlap (conflict, collision). Thus, the bus implements the half-duplex exchange mode (in both directions, but in turn, and not simultaneously).
In the bus topology, there is no central subscriber through which all information is transmitted, which increases its reliability (after all, if any center fails, the entire system controlled by this center ceases to function). Adding new subscribers to the bus is quite simple and is usually possible even during network operation. In most cases, when using a bus, a minimum amount of connecting cable is needed compared to other topologies. True, you need to take into account that two cables are suitable for each computer (except for the two extreme ones), which is not always convenient.
Because the resolution of possible conflicts in this case falls on the network equipment of each individual subscriber, the network adapter equipment with the bus topology is more difficult than with another topology. However, due to the widespread use of networks with a bus topology (Ethernet, Arcnet), the cost of network equipment is not too high.
The bus is not subject to terrible failures of individual computers, because all other computers on the network can continue to communicate normally. It may seem that the bus is not terrible and the cable is cut, since in this case we are obsessed with two fully functional buses. However, due to the peculiarities of the propagation of electrical signals along long communication lines, it is necessary to provide for the inclusion of special devices at the ends of the bus - terminators shown in Fig. 1 in the form of rectangles. Without terminators enabled, the signal is reflected from the end of the line and distorted so that communication over the network becomes impossible. So if the cable breaks or is damaged, the communication line is not coordinated, and the exchange stops even between those computers that remain connected to each other. A short circuit at any point in the bus cable disables the entire network. Any failure of network equipment on the bus is very difficult to localize, because all the adapters are connected in parallel, and it is not so easy to understand which one has failed.
When passing through a communication line of a network with a "bus" topology, information signals are attenuated and do not resume in any way, which imposes strict restrictions on the total length of communication lines, in addition, each subscriber can receive signals of different levels from the network depending on the distance to the transfer subscriber. This puts forward additional requirements for the receiving nodes of network equipment. To increase the length of a network with a bus topology, several segments (each of which is a bus) are often used, interconnected using special signal updaters - repeaters.
However, such an increase in the length of the network cannot last indefinitely, because there are also limitations associated with the finite speed of signal propagation along communication lines.

Topology "Star" is a topology with a clearly defined center to which all other subscribers are connected. The entire exchange of information takes place exclusively through the central computer, which in this way bears a very large load, therefore it cannot do anything other than the network. It is clear that the network equipment of the central subscriber must be significantly more complex than the equipment of peripheral subscribers. In this case, there is no need to talk about the equality of subscribers. As a rule, it is the central computer that is the most powerful, and it is on it that all the functions of managing the exchange are assigned. In principle, no conflicts in a network with a star topology are possible, because management is completely centralized, there is no reason to conflict.
If we talk about the resistance of a star to computer failures, then the failure of a peripheral computer does not affect the functioning of the part of the network that remains, but any failure of the central computer makes the network completely inoperable. Therefore, special measures should be taken to improve the reliability of the central computer and its network equipment. A break in any cable or a short circuit in it with a star topology disrupts the exchange with only one computer, and all other computers can continue to work normally.
On the declination from the bus, in a star on each communication line there are only two subscribers: the central one and one of the peripheral ones. Most often, two communication lines are used to connect them, each of which transmits information in only one direction. Thus, there is only one receiver and one transmitter on each link. All this greatly simplifies the network setup in comparison with the bus and saves from the need to use additional external terminators. The problem of attenuation of signals in the communication line is also solved in the "star" easier than in the "bus", because each receiver always receives a signal of the same level. A serious drawback of the star topology is the strict limitation of the number of subscribers. Typically, the central subscriber can serve no more than 8-16 peripheral subscribers. If within these limits the connection of new subscribers is quite simple, then if they are exceeded, it is simply impossible. True, sometimes a star provides for the possibility of building up, that is, connecting another central subscriber instead of one of the peripheral subscribers (as a result, a topology of several interconnected stars comes out).
The star shown in Fig. 2 is called an active or true star. There is also a topology called a passive star, which only looks like a star (Fig. 4). At this time, it is much more common than an active star. Suffice it to say that it is used in the most popular Ethernet network today.

Rice. 4. Topology "passive star"

The center of a network with this topology does not contain a computer, but a hub, or hub, which performs the same function as a repeater. It resumes the signals that are coming in and forwards them to other links. Although the cabling scheme is similar to a real or active star, we are actually dealing with a bus topology, because information from each computer is simultaneously transmitted to all other computers, and there is no central subscriber. Naturally, a passive star is more expensive than a conventional bus, because in this case a hub is also required. However, it provides a number of additional features related to the benefits of a star. That is why recently the passive star is increasingly replacing the real star, which is considered an unpromising topology.
It is also possible to single out an intermediate type of topology between an active and a passive star. In this case, the hub not only relays signals, but also controls the exchange, but does not participate in the exchange itself.
big star advantage(both active and passive) lies in the fact that all connection points are collected in one place. This makes it easy to monitor network operation, localize network faults by simply disconnecting certain subscribers from the center (which is impossible, for example, in the case of a bus), and also restrict unauthorized persons' access to vital connection points for the network. In the case of a star, each peripheral subscriber can be approached by either one cable (which transmits in both directions) or two cables (each of them transmits in one direction), the second situation being more common. A common disadvantage for the entire star topology is much more than with other topologies, the cost of the cable. For example, if computers are arranged in a single line (as in Figure 1), then when choosing a star topology, you will need several times more cable than with a bus topology. This can significantly affect the cost of the entire network as a whole.

Topology "Ring"- this is a topology in which each computer is connected by communication lines with only two others: from one it only receives information, and only transmits to the other. On each communication line, as in the case of a star, only one transmitter and one receiver operate. This eliminates the need for external terminators. An important feature of the ring is that each computer retransmits (resumes) the signal, that is, it acts as a repeater, so the attenuation of the signal in the entire ring does not matter, only the attenuation between neighboring computers of the ring is important. In this case, there is no clearly defined center; all computers can be the same. However, quite often a special subscriber is allocated in the sprat, which manages the exchange or controls the exchange. It is clear that the presence of such a control subscriber reduces the reliability of the network, because its failure immediately paralyzes the entire exchange.
Strictly speaking, sprat computers are not completely equal (unlike, for example, bus topology). Some of them necessarily receive information from the computer that is transmitting at this moment, earlier, and others later. It is on this feature of the topology that the network exchange control methods are built, specially designed for the "ring". In these methods, the right to the next transfer (or, as they say, to capture the network) passes sequentially to the next computer in a circle.
Connecting new subscribers to the "ring" is usually completely painless, although it requires the obligatory shutdown of the entire network for the duration of the connection. As in the case of the bus topology, the maximum number of subscribers in a sprat can be quite large (up to a thousand or more). The ring topology is usually the most resistant to congestion, it provides reliable operation with the largest flows of information transmitted over the network, because it usually does not have conflicts (unlike a bus) and there is no central subscriber (unlike a star) .
Because the signal in the sprat passes through all the computers on the network, the failure of at least one of them (or its network installation) disrupts the operation of the entire network as a whole. Likewise, any open or short circuit in each of the cables in the ring renders the entire network unusable. The ring is most vulnerable to cable damage, so this topology usually provides for laying two (or more) parallel communication lines, one of which is in reserve.
At the same time, the great advantage of the ring lies in the fact that the relaying of signals by each subscriber can significantly increase the size of the entire network as a whole (sometimes up to several tens of kilometers). The ring with respect to this is significantly superior to any other topology.

disadvantage ring (in comparison with a star), we can assume that two cables must be connected to each computer on the network.

Sometimes the ring topology is based on two ring links that carry information in opposite directions. The purpose of such a solution is to increase (ideally twice) the speed of information transfer. In addition, if one of the cables is damaged, the network can work with another cable (however, the maximum speed will decrease).
In addition to the three main, basic topologies considered, the network topology is also often used. tree "(tree), which can be considered as a combination of several stars. As in the case of a star, a tree can be active, or real (Fig. 5), and passive (Fig. 6). With an active tree, central computers are located in the centers of combining several communication lines, and with a passive tree - concentrators (hubs).

Rice. 5. Topology "active tree"

Rice. 6. Topology "passive tree". K - concentrators

A combined topology is also used quite often, for example, star bus, star ring.

Significance of the concept of topology.

The topology of the network determines not only the physical location of computers, but, much more importantly, the nature of the connections between them, the characteristics of the propagation of signals over the network. It is the nature of the connections that determines the degree of network fault tolerance, the required complexity of network equipment, the most appropriate method of exchange control, the possible types of transmission media (communication channels), the allowable network size (the length of communication lines and the number of subscribers), the need for electrical coordination, and much more.
When the network topology is mentioned in the literature, four very different concepts can be meant that refer to different levels of the network architecture:

1. Physical topology (i.e. layout of computers and cabling). In this content, for example, a passive star is no different from an active star, which is why it is often called simply a "star".

2. Logical topology (that is, the structure of connections, the nature of the propagation of signals through the network). This is probably the most correct definition of topology.

3. Topology of exchange control (that is, the principle and sequence of transferring the right to delight the network between individual computers).

4. Information topology (that is, the direction of the flow of information transmitted over the network).

For example, a network with a physical and logical topology "bus" can use the handover of the network capture right as a control method (that is, be a ring in this content) and simultaneously transmit all information through one dedicated computer (be a star in this content).

A computer network can be divided into two components. A physical computer network is, first of all, equipment. That is, all the required cables and adapters connected to computers, hubs, switches, printers, and so on. Everything that should work in a common network.

The second component of a computer network is the logical network. This is the principle of connecting a number of computers and the necessary equipment into a single system (the so-called computer network topology). This concept is more applicable to local networks. It is the chosen topology for connecting a number of computers that will affect the required equipment, the reliability of the network, the possibility of its expansion, and the cost of work. Now the most widely used types of computer network topologies are ring, star, and bus. The latter, however, has almost gone out of use.

"Star", "ring" and "bus" are the basic topologies of computer networks.

"Star"

Topology of computer networks "star" - a structure, the center of which is a switching device. All computers are connected to it by separate lines.

The switching device can be a hub, that is, a HUB, or a switch. This topology is also called a "passive star". If the switching device is another computer or server, then the topology can be called an "active star". It is on the switching device that the signal from each computer is received, processed and sent to other connected computers.

This topology has a number of advantages. The undoubted advantage is that the computers do not depend on each other. If one of them fails, the network itself remains operational. You can also easily connect a new computer to such a network. When new equipment is connected, the rest of the network elements will continue to operate as usual. In this kind of network topology, it is easy to find faults. Perhaps one of the main advantages of the "star" is its high performance.

However, with all the advantages, this type of computer network also has disadvantages. If the central switching device fails, the entire network will stop working. It has restrictions on connected workstations. There cannot be more than the number of ports available on the switch device. And the last disadvantage of the network is its cost. A large enough amount of cable is required to connect each computer.

"Ring"

The topology of computer networks "ring" has no structural center. Here, all workstations, together with the server, are united in a vicious circle. In this system, the signal moves sequentially from right to left in a circle. All computers are repeaters, thanks to which the marker signal is maintained and transmitted further until it reaches the recipient.

This type of topology also has both advantages and disadvantages. The main advantage is that the operation of the computer network remains stable even with heavy workload. This type of network is very easy to install and requires a minimum amount of additional equipment.

Unlike the star topology, the ring topology can lead to the paralysis of the entire system if any connected computer fails. Moreover, it will be much more difficult to identify a malfunction. Despite the easy installation of this network option, its configuration is quite complicated, it requires certain skills. Another disadvantage of this topology is the need to suspend the entire network to connect new equipment.

"Tire"

The topology of computer networks "bus" is now less and less common. It consists of a single long highway to which all computers are connected.

In this system, as in others, the data is sent along with the recipient's address. All computers receive the signal, but it receives it directly to the addressee. Bus-connected workstations cannot send data packets at the same time. While one of the computers performs this action, the rest are waiting for their turn. Signals move along the line in both directions, but when they reach the end, they are reflected and superimposed on each other, threatening the coordinated work of the entire system. There are special devices - terminators designed to dampen signals. They are installed at the ends of the highway.

The advantages of the "bus" topology include the fact that such a network is installed and configured quickly enough. In addition, its installation will be quite cheap. If one of the computers fails, the network will continue to operate normally. Connecting new equipment can be done in working order. The network will function.

If the central cable is damaged or one of the terminators stops working, this will lead to a shutdown of the entire network. Finding a fault in such a topology is quite difficult. An increase in the number of workstations reduces network performance and also leads to delays in the transmission of information.

Derived computer network topologies

The classification of computer networks by topology is not limited to three basic options. There are also such types of topologies as "line", "double ring", "mesh topology", "tree", "lattice", "close network", "snowflake", "full-connected topology". All of them are derived from the base ones. Let's consider some options.

Poor topologies

In a fully meshed topology, all workstations are connected to each other. Such a system is quite cumbersome and inefficient. It is required to allocate a line for each pair of computers. This topology is used only in multi-machine complexes.

The mesh topology is, in fact, a stripped-down version of the fully meshed topology. Here, too, all computers are connected to each other by separate lines.

Most Efficient Topologies

The topology of building computer networks called "snowflake" is a truncated version of the "star". Here, hubs connected to each other in a “star” type act as workstations. This topology option is considered one of the most optimal for large local and wide area networks.

As a rule, in large local networks, as well as in global networks, there are a huge number of subnets built on different types of topologies. This type is called mixed. Here, at the same time, one can single out both the “star”, and the “tire”, and the “ring”.

So, in the above article, all the main available topologies of computer networks used in local and global networks, their variations, advantages and disadvantages were considered.