Here, We provided to Topology Hand Written Note By P Kalika. In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing. Free download PDF Topology Hand Written Note By P Kalika.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies.
A property that is invariant under such deformations is a topological property. Basic examples of topological properties are the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles. Free download PDF Topology Hand Written Note By P Kalika.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometric situs and analysis situs. Leonhard Euler’s Seven Bridges of Königsberg problem and polyhedron formula are arguably the field’s first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. Free download PDF Topology Hand Written Note By P Kalika.
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle, the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus, and the set of all possible positions of the hour, minute, and second hands have taken together are topologically equivalent to a three-dimensional object.
BOOK INFO
BOOK NAME – TOPOLOGY HAND WRITTEN NOTE
AUTHOR – P KALIKA
SIZE – 23.7MB
PAGES – 91
Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called “rubber-sheet geometry” because the objects can be stretched and contracted like rubber, but cannot be broken. For example, a square can be deformed into a circle without breaking it, but figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from figure 8. Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900.
The following are some of the subfields of topology:
General Topology or Point Set Topology:
General topology normally considers local properties of spaces and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.
Combinatorial Topology:
Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. This is the number (V – E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have a Euler characteristic of 2.
Algebraic Topology:
Algebraic topology also considers the global properties of spaces and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated with each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups. Algebraic topology sometimes uses the combinatorial structure of space to calculate the various groups associated with that space.
Differential Topology:
Differential topology considers spaces with some kind of smoothness associated with each point. In this case, the square and the circle would not be smoothly (or differentiably) equivalent to each other. Differential topology is useful for studying properties of vector fields, such as magnetic or electric fields.
Topology Hand Written Note By P Kalika is very helpful for the aspirants of CSIR UGC NET Mathematics, IIT JAM Mathematics, GATE mathematics, NBHM, TIFR, and all different tests with similar syllabus. Topology Hand Written Note By P Kalika is designed for the students who are making ready for numerous national degree aggressive examinations and additionally evokes to go into Ph. D. Applications by using manner of qualifying the numerous the front examination. Free download PDF Topology Hand Written Note By P Kalika.
Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics, and for describing the space-time structure of the universe. Free download PDF Topology Hand Written Note By P Kalika.
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