TopologyMathematics is such a vast and diverse field that it is impossible to be an expert in all areas. It is also a field that is constantly evolving and branching outward. The field of topology is one of the most recent and intensively studied branches of mathematics. “A simple way to describe topology is as rubber sheet geometry” [2]. “Topology is a branch of geometry that originated in the 19th century and which studies the properties that an object retains under deformation, in particular bending, elongation and compression, but not breaking or tearing” [1] . Under these conditions, one could say that a square is topologically equivalent to a circle because a square can be folded and stretched into a circle [3]. However, a square is not topologically equivalent to a torus because a torus cannot be formed unless a hole is drilled in the middle or two pieces are joined together. Topologists have obviously expanded these simple concepts over time to create theorems further removed from our ordinary experiences. Some of these shapes and objects exist in four-dimensional space or higher dimensions and cannot exist in our world. Theoretically these shapes would be as common as a tree or a rock in a higher dimensional universe. However, in our universe topologists turn to mathematics to understand these shapes [6]. The first mathematical problem, which led to the origins of topology, was the Konigsberg bridge problem. The inhabitants of Königsberg wondered whether they could walk around the city so that they could cross each bridge exactly once. The map of the city looked like this [2]:Euler determined that it was indeed impossible to achieve this feat. He rationalized this problem... half of the document... trailing space. Works Cited[1] http://www.britannica.com/bcom/eb/article/2/0,5716,115452 +1,00.html EnciclopediaBritannica: Topology. Accessed December 6, 1999.[2] http://www.forum.swarthmore.edu/~isaac/problems/bridges1.html The beginnings of topology. Accessed December 6, 1999.[3] http://www.geom.umn.edu/docs/doyle/mpls/handouts/node13.html Topology. Accessed December 6, 1999.[4] http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/topology_in_mathematics.htmlTopology enters mathematics. Accessed December 6, 1999.[5] http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html FelixChristian Klein. Accessed December 7, 1999.[6] http://www.pepperdine.edu/seaver/natsci/faculty/kiga/topology.htm What is topology. Accessed December 7, 1999.[7] Yaglom, I. M. Felix Klein, and Sophus Lie. Birkhauser, Boston. 1988.