RE: [obm-l] Qual o intuito da topologia?

[Leandro Lacorte Recôva <lrecova@xxxxxxxxxxx>]

No site http://www.math.wayne.edu/%7Errb/topology.html voce tem as
seguintes colocacoes: 


Basically, topology is the modern version of geometry, the study of all
different sorts of spaces. The thing that distinguishes different kinds
of geometry from each other (including topology here as a kind of
geometry) is in the kinds of transformations that are allowed before you
really consider something changed. (This point of view was first
suggested by Felix Klein, a famous German mathematician of the late 1800
and early 1900's.) 

In ordinary Euclidean geometry, you can move things around and flip them
over, but you can't stretch or bend them. This is called "congruence" in
geometry class. Two things are congruent if you can lay one on top of
the other in such a way that they exactly match. 
......

Topology is almost the most basic form of geometry there is. It is used
in nearly all branches of mathematics in one form or another. There is
an even more basic form of geometry called homotopy theory, which is
what I actually study most of the time. We use topology to describe
homotopy, but in homotopy theory we allow so many different
transformations that the result is more like algebra than like topology.
This turns out to be convenient though, because once it is a kind of
algebra, you can do calculations, and really sort things out! And,
surprisingly, many things depend only on this more basic structure
(homotopy type), rather than on the topological type of the space, so
the calculations turn out to be quite useful in solving problems in
geometry of many sorts. 
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