Topology :: Math Mathmatics

analysis situs Topology is a modern branch of geometry. It has been called soft geometry because instead of thinking about the traditional characteristics of an object (like angles, length, etc.), topologists study features that butt endt be altered by stretching, twisting or shrinking the object. After any alteration all points in the object that were connected must still be connected and all points separated by a hole must remain separated. Topology also attempts to apologize objects that commodenot exist in three attributes using mathematical equations, since it is nearly out of the question to imagine such objects within our frame of reference. The dimension of an object shtup be thought of in two ways intrinsic and extrinsic. The erudition of a creature occupying, say a line, is one-dimensional, since he can hardly move in one dimension. However, we draw a line on a plane, so extrinsically it is two-dimensional (1). So how do objects occupying the same dimension differ topologically? A doughnut shaped object, called a torus, and a field of operation atomic number 18 topologically different. Both of these objects are extrinsically two-dimensional, since we only deal with the surfaces of the object. There is no inside. The rea male child for the topological difference is the hole in the middle of the torus. No permitted alterations (stretching, twisting, shrinking) can be made to the sphere that will transform into a torus. Topology emerged out of Eulers work on graph theory in the early 1700s. Leonhard Euler was born on April 15, 1707 in Switzerland. His father was a see and wanted his son to follow in his footsteps. He sent his son to the University of Basel in 1720, when Leonhard was only 14. It was here that his interest and natural capabilities in mathematics really began to show. After completing his studies and showing very promising mathematical talent, Euler moved to St. Petersburg, Russia to t each mathematics, at the age of only 19. He remained in Russia for several years (4). And it was here that he made contributions to mathematics that would subsequent be seen as the first steps towards topology. Graph theory studies how points are connected without giving any regard to the distance between them or the actual shape of the line connecting them.