Topology

Topology is a branch of mathematics that studies the characteristics of geometric objects that are retained during constant deformations including stretching, crumpling, twisting, and bending.

A topological space is a collection with a topology that allows for the definition of continuous deformation of subspaces and, more broadly, all other forms of continuity.

Any distance or metric determines a topology, so Euclidean spaces and, more broadly, metric spaces are representations of topological spaces. Homeomorphisms and homotopies are two types of deformations that are taken into account in topology meaning. A topological property is one that is invariant under this kind of deformations.

The dimension that distinguishes between a line and a surface; connectedness, between a circle from two non-intersecting circles, compactness, and between a line and a circle; are all instances of topological properties.

Motivation

Topology is motivated by the realization that certain geometric problems are determined not by the exact form of the objects concerned, but by how they are placed together. The square and the circle, for instance, share several characteristics, such as, they both are one-dimensional objects (from a topological standpoint) and they divide the space into two sections, one within and one outside.

Leonhard Euler showed in amongst the first papers in topology that it had been difficult to find a path through Königsberg (now Kaliningrad) which would pass each of its seven bridges precisely once. This result was independent of bridge lengths or distances from one another, but only of connectivity properties, such as which bridges link to which islands or riverbanks.

Topology and its Types

Types of Topology

The types of topology have been mentioned below:-

• Ring Topology

• Mesh Topology

• Bus Topology

• Star Topology

• Hybrid Topology

Lan Topology

Linear bus, ring, and star are the three most popular LAN topologies. Another network topology that can include components of the bus, ring and star is hierarchical. The physical and logical topologies for a LAN (Lan topology) are determined by reliability, performance and cost goals, and also user connectivity specifications.

Wan Topology

Wide area networks, or WANs (Wan topology), are used in the workplace to link locations that are spread across the globe. A topology is a definition of how something is laid out or arranged. Applying the idea of topologies to WANs necessitates two distinct yet intertwined viewpoints.

The physical topology, which explains the physical layout of network devices that allows data to pass from a source to a destination network, is one viewpoint to understand. The logical topology, which explains how data travels around the WAN, is another viewpoint.

General Topology

The branch of topology that deals with the fundamental set-theoretic concepts and constructions used throughout topology are known as general topology. Many other branches of topology, such as geometric topology, differential topology, and algebraic topology, are built on it. Point-set topology is yet another term for general topology.

Topological spaces are collections with a topology, i.e., a collection of subsets termed as open sets which are locked within finite intersections and unions. Accessible collections can be used to describe fundamental topology principles like compactness, continuity, and connectedness. Continuous functions, on the surface, appear to connect neighbouring points.

Algebraic Topology

Algebraic topology is a mathematics branch that studies topological spaces using algebraic methods. The general objective is to find algebraic invariants which differentiate topological spaces up to homeomorphism, but most only identify topological spaces up to homotopy equivalence.

Homology, Homotopy groups, and cohomology are by far the most common invariants.

Differential Topology

Differentiable functions on differentiable manifolds are the subject of differential topology. It is strongly connected to differential geometry, and the geometric theory of variational manifolds is made up of both.

Differential topology is concerned with structure and properties that can be described using only a smooth structure on a manifold.

Geometric Topology

Geometric topology is a category of topology which works on low-dimensional manifolds (areas with measurements of 2, 3, and 4), as well as their interactions with geometry. It also contains a certain higher-dimensional topology. Orientability, crumpling, local flatness, handle decompositions, and the planar and higher-dimensional Schönflies theorem were several instances of geometric topology topics.

Generalizations

On occasion, topology tools are needed but a "collection of points" is not available. The lattice of open sets is the core idea in useless topology, whereas Grothendieck topologies were frameworks defined on arbitrary classes which enable the description of sheaves on such classes, and thus the description of general cohomology theories.

Network Topology and its Types

Network topology refers to the physical as well as logical configuration, virtual form, or structure of a network. At around the same time, a network may have a single physical topology and multiple logical topologies. There are basically two types of network topologies, namely, Physical Topology and Logical Topology.

• Physical Topology: A physical topology defines how computers or nodes in a computer network are linked to one another. It is the configuration of various elements (links, nodes, etc.) of a computer network, such as system placement and code installation. In other terms, it is the physical configuration of the network's nodes, workstations, and cables.

• Logical Topology: The way data flows from one machine to another is defined by logical topology. It is linked to a network protocol that specifies how data is transferred across the network and along which direction it travels. To put it another way, it's how the computers interact internally.