Think of a surface as infinitely stretchable and infinitely thin.

In 3 dimensions, topologists don't study solids (i.e., cubes, pyramids, spheres, etc.) the way that geometers do. They are interested in the **surfaces** of these and many other objects, real and imaginary. The surface of a cube, a pyramid, and a sphere are topologically equivalent. A stretchable "skin" that covers any one of them can be restretched to cover any of the others. The surface of a donut and a coffee cup are **topologically
equivalent**--each is a three-dimensional object with a hole in it. The surface of a sphere and a donut are not, however, topologically equivalent. This is an unusual (but valid) way to think about the world.

Topologists don't limit themselves to the three dimensional world with which we are familiar. Many of the concepts and theorems in topology deal with many-dimensional surfaces which exist only in the imagination. What do you think surfaces in 4 dimensions would be like?

Topologists are interested in a variety of surfaces. They try to understand exactly what it is that distinguishes one surface from another, and to understand the relationships that a surface can have with the space that it occupies. The mathematical study of knots is a branch of topology. The idea of a Seifert Surface also comes from Knot Theory (with a little bit of bubble-blowing thrown in!)