Classification

We can make objects like the Möbius Band by taking a strip of paper and gluing the ends together with a twist of an odd multiple (positive or negative) of 180 degrees.

There is no way you can move one of these to another in our three dimensional space.

Yet from another point of view, these objects are all the same! You can match one to another in a continuous way by matching the fibres as you move round the two circles - because the gluing is with a twist of an odd multiple of 180 degrees, these matchings of fibres themselves match up to give a kind of abstract way, a kind of rule or law, which tells you how to transform the points of one to the points of the other in a continuous way. In this sense, there is only one Möbius band. On the other hand, this matching can be accomplished by movements only in four dimensions. We have not found out how to illustrate this!

There are similar questions of classification for other types of fibre bundles. A basic question is that of the kind of properties you wish to preserve when you say two objects 'are the same'.  

How to classify objects and how to list your classes of these objects is a basic problem in mathematics and science. This problem is discussed elsewhere in the context of Knots and Links in the Knot Exhibition..