This object has only one surface and one edge, as can be seen by coloring the middle of the strip or the edge with a crayon. But topologists are also interested in the properties of objects that cannot be visualized in any usual sense.
As in other areas of mathematics, topologists try to be as general as possible in drawing conclusions. Often they do not restrict themselves to the two- or three-dimensional space of experience, but ask about the characteristics of objects in four, or five, or 500 dimensions. These studies are not necessarily sheer flights of fancy. An equation in five variables defines a surface or "hypersurface" in five-dimensional space. Topologists will want to know whether the surface is closed or infinite in extent, and about how curved different parts of the surface might be.
Differential topology is the study of the curvature of generalized surfaces, or, as topologists call them, manifolds. Measuring the curvature of a surface in a space of more than three dimensions is difficult and topologists often deal with the problem by taking what might be called an "ant's eye view." Suppose that a mathematically inclined ant is walking over the surface of a very large mound.
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