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TopologyForBeginners? needs to be written. :-) Or is such a thing even possible?

Topology is a branch of mathematics dealing exclusively with properties of continuity. Formally, a topology for a SeT 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.

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.
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...

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.

As far as 2-manifolds in three space go, you can embed any of them except for non-orientable surface without boundaries. That leaves the sphere with any number of handles and holes, the MoebiusBand? with any number of handles and extra holes, and a pair of Moebiusbands glued partway edge-to-edge with any number of handles and extra holes. This sort of thing should go on ManiFold.

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.

which exercise is that? if this is the enclyopedia page on topology (the initial page, anyway, which should expand ultimately into many), the exercise is not primarily about modelling the physical world, but about doing mathematics

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?

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.

When one is studying the topology of surfaces, there is no escaping the fact that surfaces are 2-dimensional. A torus's surface is not 2-dimensional and a torus does not have guts as part of its identity, because a torus is a surface. That we cannot build these objects in our physical 3-space because of physical limitations is irrelevant. We cannot exhibit a straight line in our 3-space either, or a flat surface, or even a point for that matter. These are all mathematical objects and this is an encyclopedia page about mathematics.

However, it is useful to point out that we can build a Mobius strip exactly in 3-space. That is, we can exhibit a set of points in 3-space which is homeomorphic to any and all Mobius strips. That is a triviality. However, we cannot build a Klein bottle in 3-space. So there really is an significant and purely mathematical difference here.

Side note: Mobius is actually spelt with an umlaut, two dots over the o. The preferred anglicization is to change this to an oe, but sometimes people just drop the o. It's not so much an incorrect spelling as a non-conventional one. (Of course "Mobious" is not an accepted spelling under either scheme.)
"Mobious" was a typo. Anything that is a typo in something unsigned can pretty safely be edited out. In fact, if this is to be a useful resource for anyone, the whole page should be radically edited. Among possible pages of a similar length, the average Joe on the street would not get very much edification out of it.

The discussion of the non-existence in our physical universe of Mobius strips (as well as lines, planes, and every other object studied in mathematics) should then be placed in a separate page that discusses mathematical ontology in relation to physical ontology, or something along those lines, since it comes up in discussion of more than one sub-discipline of mathematics, although not very much in the practice of mathematics proper.

-- CalvinOstrum

Better yet the discussion of the non-existence in our physical universe of Moebius (Mobius, whatever) strips (and everything else) could be edited out altogether. It adds very little substantive value to the understanding of any branch of mathematics, certainly not topology as I understand it.

Just my 2 cents.

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Last edited February 13, 2001 2:49 am by cobrand.bomis.com (diff)