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					                                            Topology



Webster’s dictionary defines topology as “a branch of mathematics concerned with those
properties of geometric configurations (as point sets) which are unaltered by elastic deformations
(as a stretching or a twisting) that are homeomorphisms” (1) Although the ideas of topology have
been around for a couple of hundred years, topology is a relatively new field of study. Euler is
considered the first person to start dealing with topology. He worked with a geometry where
distance did not matter. This was in 1736 that his papers were published. A.L. Cauchy also
studied topology. He studied the n-dimensional “hyperspace” as did Arthur Cayley and
Hermann Grassmann. In the late 1800's, Georg Cantor showed “that the points of geometrical
figures like squares, “clearly 2-dimensional”, could be put into one-to-one correspondence with
the points of straight line segments, “obviously 1-dimensional”.”(2) From his studies, the word
dimension started to be questioned more intensely. Cantor’s work came from his studying a way
of showing a function using a trigonometric series. He corresponded with Dedekind about his
works. Dedekind came up with an invariance theorem which Cantor accepted as better than his
own theorem. From 1880-1900, the invariance of dimension was established.
Point Set Topology was the next idea that was explored. Cantor also played a part in this
evolution. He published six papers from 1879-1883 that investigated set theory. One of
Cantor’s fundamental concepts is that of limit point.(2) Giuseppe Peano and Camille Jordan
began using Cantor’s ideas in geometry. Jordan is recognized for his theorem on closed curves
in a plane. This was published near the end of the nineteenth century. Peano created an example
of a space-filling curve. This is a curve that covers all of the points of a square. In 1906,
William H. And Grace Chisholm Young published Theory of Sets of Points. This book showed
curves that had been explored up to this point. This information became recognized to many
mathematicians of the day.
Dimension and invariance were still being investigated and as the twentieth century started, new
ideas were presented. Rene Baire, Maurice Frechet, and Frigyes Riesz all created new ideas
about dimension. Henri Poincare came up with a new idea for dimension. He is considered one
of the founders of algebraic topology. His investigations about dimensions came about as he
studied philosophy. He was trying to find the relationship between geometry and space.
Poincare’s first write-up of his theory did not answer all of the questions. He continued to work
on this topic and in 1903, he published a new version of his theory. Another update was given in
1912.
Another key person in the development of topology was Juitzen Egbertus Jan Brouwer.
Brouwer’s ideas were from a different standpoint that Poincare’s work. L.E.J. Brouwer looked
at topology using mapping, degree and dimension. He published papers between 1909 and 1913
on these topics. These ideas were not accepted by all. Korteweg was one who was not
completely happy with Brouwer’s works. Since Korteweg was Brouwer’s teacher, he changed
his path of research for a while to try to help improve his standings in the mathematical realm.
He then produced more than 40 papers in less than five years on topology and continuous group
theory. (2) Around 1910, Brouwer published his proof for dimensional variance. In the same
publication, Henri Lebesgue also published the same results. Brouwer challenged Lebesgue to
show all of his work to back up the results. Brouwer figured out the proof himself after
Lebesgue did not show his work. Poincare’s work on dimension was studied by Brouwer who
decided that it was not good enough. Brouwer published another paper in 1913 in which he
showed his proof of dimensional invariance as well as his proof of Lebesgue’s problem. In
1921, Lebesgue finally published a proof of his results. It took him ten years to get this
accomplished and he did not find the success that some of the other fathers of topology saw.
In 1914, Felix Hausdorff published Fundamentals of Set Theory . He dealt with set theory with
topological spaces. Hausdorff also put together ideas from Hilbert, Frechet and Weyl. This book
was easy to read, but did not include a definition of the concept of a curve. He continued his
works and published a theory dealing with metrical dimension instead of topological dimension.
Pavel Urysohn, started Moscow University in 1915 and by 1921 he began to study topology.
Within one year his theory of dimensional topology was just about finished. His theory was
expanded by others during the 1920's as they searched to find answers to other questions about
topology. Urysohn died in 1924 while swimming in the sea off of the coast of Brittany. Karl
Menger attended a meeting about curves and dimension and wrote a paper where he tried to
define a curve in Euclidean 3-space. He then dealt with 1-dimension and how curve theory
works with dimension theory. By 1926, Menger published his dimension theory. In the 1930's,
new types of topology began to be established. They included homology theory and algebraic
topology.
One common story in topology is about a coffee cup and a donut. Two spaces are topologically
equivalent if one can be transformed into the second without actually taking it apart. In theory a
donut could shaped into a coffee cup by making an indentation in the side of the donut and
gradually making it larger while shrinking the hole into the handle. This applies to the theory of
topological equivalency. Topology is a relatively new field and continued to evolve and to be
created into the middle of the twentieth century.


Sources:
1. www.m-w.com
2. History of Topology, I.M. James, 1999
3. http://www-history.mcs.st-andrews.ac.uk
4. http://en.wikipedia.org/wiki/Main_Page

by: Tabitha Tutterrow

				
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