Two objects are topologically equivalent if one object can be continuously deformed to the other. In one or two dimensions, this is something we can visualize: to continuously deform a surface means to stretch it, bend it, shrink it, expand it, etc.--anything that we can do without actually tearing the surface or gluing parts of it together. Intuitively, we imagine that the surface is made of infinitely flexible rubber, and any shape that we can transform this surface into without tearing or gluing the rubber is topologically equivalent to it.

In two dimensions, there are many examples of surfaces that are topologically equivalent, and also many examples of surfaces that are topologically distinct (i.e., surfaces that cannot be continuously deformed to each other). To begin with, consider an ordinary piece of paper, without the edges (that is, the edges of the paper are not...