The crucial problem in topology is deciding when two shapes are equivalent. Unlike Euclidean geometry, which focuses on the measurement of distances between points on a shape, topology focuses on the similarity and continuity of certain features of geometrical shapes. For example, in Figure 1, each of the two shapes has five points: a through e. The sequence of the points does not change from shape 1 to shape 2, even though the distance between the points, for example, between points b and d, changes significantly because shape 2 has been stretched. Thus the two shapes in Figure 1 are topologically equivalent, even if their measurements are different.
Similarly, in Figure 2, each of the closed shapes is curved, but shape 3 is more circular, and shape 4 is a flattened circle, or ellipse. However, every point on shape 3 can be mapped or transposed onto shape 4.
Shapes 1 and 2 are both topologically equivalent to each other, as are shapes 3 and 4. That is, if each were a rubber band, it could be stretched or twisted into the same shape as the other without connecting or disconnecting any of its points. However, if either of the shapes in each pair is torn or cut, or if any of the points in each pair join together, then the shapes are not topologically equivalent. In Figure 3, neither of the shapes is topologically equivalent to any of the shapes in Figures 1 or 2, nor are shapes 5 and 6 equivalent to each other. The circles in shape 5 are fused; and the triangle in shape 6 has a broken line hanging from its apex.