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Mathematics-Topology

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Subject- Mathematics
Topic- Topology
University Name- Northwestern University

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Topology
Mathematics is a field so vast and diverse 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
newest intensively studied branches of mathematics. “A simple way to describe topology is as
rubber sheet geometry” [2]. “Topology is an offshoot of geometry that originated during the 19th
century and that studies those properties an object retains under deformation - specifically,
bending, stretching and squeezing, 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 bent 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 bored through the medium, or two pieces are joined
together. Topologists obviously have expanded upon 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 commonplace as a tree or 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
bridges problem. The people of Konigsberg wondered if they could walk around the city in a
way that they would also cross every bridge exactly once. The city map looked something like
this [2]:
Euler determined that it was indeed impossible to accomplish this feat. He rationalized this
problem by drawing a simpler picture. He put a dot, on each side of the river and extended lines
from the vertices to the other possible locations one could go from this point. He then counted up
the possible places one could go from the vertices. In this picture the vertices all have an odd
number of options extended from them. Euler said that this feature made the feat impossible to
accomplish because it is only possible to have two odd numbered vertices, one to start at and one
to finish at. This problem was solved and published by Euler in 1736. The name of his
publication alone, The Solution of a Problem Relating to the Geometry of Position, indicates that
this problem dealt with a new type of geometry where distances did not play a role. Eulers

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? Subject- Mathematics ? Topic-?Topology University Name- Northwestern University Topology? Mathematics is a field so vast and diverse 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 newest intensively studied branches of mathematics. "A simple way to describe topology is as rubber sheet geometry" [2]. "Topology is an offshoot of geometry that originated during the 19th century and that studies those properties an object retains under deformation - specifically, bending, stretching and squeezing, 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 bent 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 bored through the medium, or two pieces are joined together. Topologists obviously have expanded upon 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 commonplace as a tree or 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 ...
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