TopOlogy

Topology is a branch of mathematics dealing exclusively with properties of continuity.  Formally, a topology for a space X is defined a set T of subsets of X satisfying:

1) T is closed under abitrary unions


2) T is closed under finite intersections


3) X, {} are in T



The sets in T are referred to as open sets, and their complements as closed sets.  Roughly speaking open sets are thought of as neighborhoods of points.  This definition of topology is too general to be of much use and so normally additional conditions are imposed.
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The importance of TopOlogy is, in part, that one can define different topological spaces with elements from AnalySis, AlgeBra, or GeoMetry and then one can determine the properties of such spaces and prove theorems about them. Of equal importance, one can prove what is '''not''' true about such spaces. Thus, through TopOlogy one can obtain results in AnalySis, AlgeBra and GeoMetry. This makes TopOlogy very powerful.
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In lay terms, what is constant about objects of a given topology is how many continuous surfaces, spaces and boundaries can be envisioned.  A sphere and a bowl have the same topology.  So do a doughnut and a teacup, (owing the the loop forming the handle of the cup) both of which are a simple torus.  The study of topology introduced us to theoretical objects such as the Moebius strip and the Klein bottle which can be depicted, even modeled after a fashion, but not actually built in three-dimensional space.

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Just to be clear, a Moebius strip can be modeled in 3D space (we made them as kids all the time), it's the Klein bottle that cannot.

Its been a while since I've had topology, but I seem to remember there being a theorem indicating that there were only some specific number (6 is it?) of different topological shapes in 3-space.  Does anyone remember this?

I've been out of it for far too long...

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Modeled, yes, but built, no.  The Moebius strip is a theoretical object consisting of a two-dimensional planar surface limited by parallel lines which is twisted 180 degrees and joined to itself.  As there is no real, true planar existence in our three-dimensional world, we cannot build the Moebius strip.  The paper has thickness, however minor, and so is in reality a torus once joined to itself, the topological equivalent of a doughnut.

''And strictly speaking the paper is not actually continuous, but is actually composed of many separated atoms, so isn't really a torus.  I think there is a certain point of "good enough" for making pictures/sculptures of mathematical objects, and that the Moebius strip in paper meets it.''

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OK - fair enough, but the generally, the purpose of this whole exercise is to use a mathematical model to further understand something that exists, not bring into existance things have the exact characteristics of any given mathematical model.  The Klein bottle is interesting because there is nothing physical that it is modelling, but we can analyze it anyway.

As far as your "paper is not continuous" argument, does this mean that we should be using Toplogical models that have CantorSet qualities in order to keep our models more like the real world?