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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I don't know what you mean by "different topological shapes in 3-space", but I can't imagine a meaning for which there would only be 6 of them.  Even if we restrict ourselves to nice objects such as compact surfaces (which seems to be what you have in mind) there are infinitely many pairs that are non-homeomorphic. There is a representation theorem that characterizes them in very simple terms, however.

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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 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 make our models more like the real world?

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3-D Euclidean space itself is as much a "theoretical object" as a Mobius strip, and the physical analogues we build of Mobius strips are certainly closer to actual Mobius strips than to tori. Tori are ''not'' three dimensional objects -- they are two dimensional objects -- compact connected 2-manifolds.   That is, a torus is not the topological equivalent of a donut, for it is hollow.

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Fair enough.  My point initially was that such theoretical shapes as Moebius strips and Klein bottles '''can''' be '''modeled''' after a fashion but not ''actually literally'' '''built''' in our three-dimensional space.  The fact that we can model a Moebius strip is instructive, and serves to get students thinking in new ways, but the 3-dimensionality of the materials used should not ultimately be ignored.  Yes, of course a torus is a two-dimensional object, deformed through a third dimension.  I referred to the surfaces of these objects, not their "guts."  As was pointed out earlier, the nature of matter, being charged particles with space between them, means we can't actually '''build''' any of these theoretical mathematical objects which consist of sets of points with no space between them.  Here is a new thought - do you suppose that faster-than-light travel will be invented by the same guy who figures out how to construct a real Klein bottle?  Or better still, figures out how to deform our three-dimensional space through an additional dimension?