Topology - Math Assignment Help

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Topology

Topology is a branch of mathematic in which two objects can be deformed into one another through bending, twisting, stretching, shrinking etc. Topology refer to qualitative objects like coffee mug, droughnut means objects have a hole in their shape.

Various basic concepts of topology include

Simply connected means any closed curve can be shrink to a point continuously in the set means path wise connected and every map from one sphere extends continuously to a map from the second disk. Every loop in the set is contractible. Objects in topology are either three,two, one dimensional shape. For example surface of sphere is not topologically equivalent to torus and the surface of droughnut ring.

Topological equivalence- It is the relationship of two figures that can be transformed one into the other by a one to one transformation. It is reflexive, symmetric, transitive .For example consider an object circle and a point inside the circle and a point outside the circle but in the same plane or a circle is equal to sphere in shape and a sphere is equal to ellipsoid.

Homeomorphism- a function f is homeomorphism and objects X and Y are said to be homeomorphism if and only if

  • H is a one - one relation between elements of X and Y.
  • H is continuous
  • Exists an inverse of h so that for any x in X or any y in Y the original value can be restored by combining the two functions in systematic order.

It means two objects can be manifold by a continuous, invertible mapping.

Topology Structure - It is a concept in which a set X is turned into a space by assuming a collection of subsets T of X. It has three axioms

1. set X itself and the empty set are members of T

2. intersection of any finite number of sets in T is T

3. Union of any collection of sets in T is in T.

Set T are called open sets and T is known as the topology on X. For example a number line becomes a space when it is a collection of all the open intervals.

To sum up we say that it takes care of all the properties such as convergence, connectedness and continuity while doing transformation.

Algebraic topology- It is a branch of mathematic that uses concepts from abstract algebra to study spaces (topology). The goal is to find the basic goal is to find algebraic invariant associated with different structures. Example- Euler characteristic a number associated with a surface. Euler proved V-E+F=2 which relates the number V and E of vertices and edges of a network that divides the surface of a spaces and functions to the objects and the maps are known as homeomorphisms.

Different forms are as follows-

1. Fundamental group- It is a group that determines when two paths starting and finishing or ending point from a fixed point can be deformed into each other. It is known as homology groups of topological space and these are classified as homology groups.

2. Differential topology - It deals with differentiable functions on differentiable manifolds. It is related to geometry and together makes up the geometric theory of differentiable manifolds. It is a separate branch of differential topology related to calculus of variations is the global theory of extremes of various functions on manifolds of geodesics. It influenced the development of topology by making a possible a classification of vector bundles and by producing a method of studying the invariants provided by KK theory.

3. Knot Theory - It is the embedding of one topological space into another. It means ends are joined together so that it cannot be undone. For example a unit circle can be embedded into a three dimensional space. It is a closed piecewise linear curve. Two or more knots tied together are called link. Two knots or links are equal if one can be deformed into the other.

Few reasons why does the topology or shape of a network matter can be

1. It impacts performance

2. It is determining factor used to determine the media type used to cable the network.

3. Access methods can work only with specific topologies.

Distance travelled and layout of path as it moves from the sending computer to receiving computer on the network greatly impacts the speed of communication. Knowledge of the layout of your network will help you not only optimize your network's operation but also troubleshoot performance issues. For example thin coaxial cable is usually associated with the linear bus topology while unshielded pair cable is associated with the star bus topology. Type of cabling depends on type of topology.

It impacts cost in two ways

1.      Cable types are associated with different topologies vary in cost

2.      Different topology require different amount of cable.

Four most common topologies are linear Bus, Star Bus, Ring, Mesh (hybrid or partial). Topology may determine a network's media access. It is the means by which access to the network is controlled and data collisions (computer talking at the same time) are prevented. Some methods are associated with more than one topology. For example - token passing is an access method that is used with a ring topology (token ring) or a star or linear bus.

To conclude we say that Topology is an area where stretching, bending but not twisting of an object.

It is the study of qualitative properties of certain objects especially geometric configurations. It operates with more general concepts than analysis. Differential property is non-essential but bicontinuity is essential.It is apt to use where conclusions cannot be made from analysis. It is used in data mining, computer aided design, digital topology, information systems etc.  The idea of set and function in spaces are extensively developed and used in many engineering problems , information systems , particle physics and mathematics science. All attributes defined above will have many applications in topology.

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