Topology . The youngest of the classical branches of Mathematics . In contrast to Algebra , Geometry, and Number Theory , whose genealogies date back to ancient times, topology appears in the seventeenth century, under the name of analysis situs, that is, analysis of position.
“Topology is qualitative Mathematics, that is, mathematics without numbers. We can roughly say that Topology deals with the intrinsic properties of spatial configurations, which are independent of size, situation and shape … An intrinsic qualitative property it is one that does not change when the object under consideration undergoes a process of expansion and flexion without rupture ” [1]
Summary
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- 1 Story
- 1 From the name day
- 2 Features
- 3 Types of topology
- 1 Ring topology
- 2 Star topology
- 3 Bus topology
- 4 Mesh topology
- 4 References
- 5 Available literature
- 6 Sources
History
The concepts of adjacency, environment, of the infinitely close, likewise the concept of dissection of a body, to some extent interchangeable with the previous concepts, are those that Lobachevski places in the sustenance of the entire framework of geometry and are also , the liminal concepts, in synthesis, the fundamental concepts of Topology in all the breadth in which this discipline is currently understood. [2]
In anticipation of the origin of Topology, the classical definition of continuity should be taken. The more is x near to the closer of l is g (a) . Clear proposal enunciated by the Gallic mathematician, Luis Cauchy, in 1821. Date that marked, for this case and other concepts introduced by him, a revolutionary change in mathematics, which led to the development of the process that has led to the site of the enormous current expansion and application. [3]
The beginnings of Topology are found in the work of Carlos Weirstrass, by 1860, in which he analyzed the concept of the limit of a function – as used in calculation. In this effort he rearmed the real number system and made some of its now named “topological” features transparent. Next, the great foundational contribution of Jorge Cantor, member of the Academy of Sciences of Saint Peterburg (Russia) on the theory of set of points (1874-1895), establishes the foundations on which Topology would circumstantially build its own scaffolding. A second nuance of Topology, called algebraic or combinatorial topology began in 1890 with the outstanding work of Enrique Poincaré on the theory of integral calculus in poly-dimensional spaces.F. Hausdorff and others, in the period from 1900 -1910, established solid foundations for the first topological nuance, which is nothing but the so-called conjunctival topology . LEJ Brouwer, in his investigation (1908-1912) of the concept of dimension, was the pioneer in treating a mixture of conjunctivist and combinatorial aspects of Topology. The integrated theory experienced a great expansive rhythm, in the period from 1915 to 1930, as a result of the efforts of the mathematicians JW Alexander, PL Alexandrov, Solomon Lefschetz and others. Until 1930 Topology was called as in situ analysis . Lefschetz was the original author and popularizer of the word topology, having brought to light a treaty with such a title in 1930. Onomatic term of universal acceptance.
From the name day
From this date – 1930- the Topology has developed at an almost exponential rate. This discipline ventured into the calculation of variations by means of the theory of critical points forged by the American mathematician, M. Morse. A new impetus is given to differential geometry by the work of H. Whitney. Specifically, on fibrous packages, the work of G. de Rham and that of Lie groups by H. Hope open up new perspectives. A minor revolution was made in modern algebra with the deployment of new livelihoods for algebra and a new branch, known as homological algebra. Much contribution owes this product to S. Eilenberg and S. Mac Lane. Topology gave a further boost to algebraic geometry through beam theory and cohomology and produced important new applications in the theory of partial differential equations based on the work of J. Leray and M. Atiyah. The influence of Topology on differential geometry has initiated topological thinking in relativity theory. Topology has become a basic discipline of Mathematics, in practice, a necessity in some subjects (a course in Mathematical Analysis supposes an introduction on the concept of neighborhood and openness, to say, or in graph theory). Above all, an integrating energy for almost all Mathematics (highlighted in the bourbakist orientation). The influence of Topology on differential geometry has initiated topological thinking in relativity theory. Topology has become a basic discipline of Mathematics, in practice, a necessity in some subjects (a course in Mathematical Analysis supposes an introduction on the concept of neighborhood and openness, to say, or in graph theory). Above all, an integrating energy for almost all Mathematics (highlighted in the bourbakist orientation). The influence of Topology on differential geometry has initiated topological thinking in relativity theory. Topology has become a basic discipline of Mathematics, in practice, a necessity in some subjects (a course in Mathematical Analysis supposes an introduction on the concept of neighborhood and openness, to say, or in graph theory). Above all, an integrating energy for almost all Mathematics (highlighted in the bourbakist orientation).
characteristics
Topology is a branch of mathematics that studies continuity and other concepts originating from it. It is a specialization linked to the properties of geometric bodies that remain unchanged by continuous transformations, regardless of their size or shape. The continuous functions of mathematics are those that, at the points near the domain, undergo small variations in values. At the graphic level, these functions are usually in a position to be drawn without the need to lift the pencil from the paper. Another central concept of topology is the topological space, which is a mathematical structure that allows the formal definition of continuity, connectivity and convergence, among other concepts.
Topology, therefore, is the specialization that studies continuous functions and topological spaces. This discipline works with objects in different ways, provided that the mentioned continuity is not interrupted. In the words of everyday language, it could be said that topology is allowed to bend, stretch, twist or shrink objects, but without breaking them or separating what was united (or pasting what was separated). At the topological level, a triangle is the same as a circumference: one can be transformed into the other continuously, without the need to cut or paste. Instead, a circumference can never be transformed into a segment from the topological point of view, since such transformation would require breaking the continuity of the figure. Among the branches of topology,
Topology types
Topology defines the structure of a network. The topology definition is made up of two parts, the physical topology, which is the actual arrangement of cables (the media), and the logical topology, which defines how the hosts access the media. The commonly used physical topologies are bus , ring , star , extended star, hierarchical, and mesh .
Ring topology
A ring topology consists of a single closed ring made up of nodes and links, in which each node is connected only to the two adjacent nodes.
The devices connect directly to each other using cables in what is called a daisy chain. In order for the information to circulate, each station must transfer the information to the adjacent station (all hosts are connected to each other forming a closed-loop chain.)
Star Topology
It has a central node from which all links are radiated to the other nodes. All the information circulating on the network passes through the central node, generally occupied by a hub. The main advantage is that it allows all nodes to communicate with each other in a convenient way. The main disadvantage is that if the central node fails, the entire network goes offline.
Bus topology
The bus topology has all of its nodes connected directly to one link and has no other connection between nodes. Physically, each host is connected to a common cable, so they can communicate directly, although the cable break causes the hosts to be disconnected. The bus topology allows all devices on the network to see all signals from all other devices, which can be advantageous if you want all devices to get this information. However, it can be a disadvantage, as it is common for traffic problems and collisions to occur, which can be mitigated by segmenting the network into several parts. It is the most common topology in small LANs, with hub or switch. end at one end.
Mesh topology
Each node is directly linked to the other nodes. The advantages are that, as each everything is physically connected to the others, creating a redundant connection, if any link stops working, the information can circulate through any number of links until it reaches its destination. Furthermore, this topology allows information to circulate on various routes through the network. The main physical disadvantage is that it only works with a small number of nodes, since otherwise the amount of media required for the links, and the number of connections to the links becomes overwhelming.
