Möbius is called a "pioneer" of topology because he was one of the first mathematicians to develop it. Topology is a branch of geometry that deals with specific properties of geometric shapes that stay the same, no matter how much the shapes are bent, twisted, stretched, or deformed. The only rule is that distinct points in the space cannot coincide and that "tearing" the space cannot be allowed. Topology's nickname is rubber-band geometry. Geometry is concerned with properties such as absolute position, distance, and parallel lines, but topology is more concerned with relative position and general shape.

Encarta's topology article explains, "For example, a circle divides a flat plane into two regions, an inside and an outside. A point outside the circle cannot be connected to a point inside by a continuous path lying in the plane without crossing the circle. If the plane is deformed, it may no longer be flat or smooth, and the circle may become a crinkly curve; it will, however, maintain the property of dividing the surface into an inside and an outside. Straightness and linear and angular measure of the plane are some of the properties that are obviously not maintained if the plane is distorted."

An early topology problem is the Königsberg bridge problem: Is it possible to cross the seven bridges over the Pregel River, connecting two islands and the mainland, without crossing over any bridge twice?

The Swiss mathematician Leonard Euler showed that the problem was the same as this: Is it possible to draw this figure without lifting pencil from paper, and without tracing any edge twice?

After turning the bridge problem into a diagram, Euler was able to prove that it was not possible. Euler went on to prove that any connected linear figure may be drawn with one continuous stroke without retracing edges if and only if the figure has either no odd vertices or just two odd vertices (a vertice is odd if it is the endpoint of an odd number of lines).

Later, the German mathematician Johann Benedict Listing proved that a connected linear graph with 2n odd vertices can be drawn with n continuous strokes, each starting and ending at an odd vertex.

Today, topology still is an active branch of mathematics. Map-makers have always wondered how many colors are needed so that no two bordering regions have the same color. In 1976, Kenneth Appel and Wolfgang Haken used topology and a computer to prove that just four colors are enough.

Knots also are part of topology - the goal is to unravel the knot by stretching and pulling, but you can't tear. In knot theory, two knots are the same if they can be stretched into the same shape. Therefore, figures 4a and 4c are the same, but figures 4a and 4b are different. Knot theory is a part of topology that still has many unsolved problems.