# Topology

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### What is Topology?

Topology is a branch of pure mathematics that deals with the abstract relationships found in geometry and analysis. The word "topology" means the study of surfaces. Topologists attempt to understand shape and space without an explicit measure of distance or size. You may hear topology described as "rubber sheet" geometry. That is, if geometric figures are drawn on a rubber sheet, then stretched and contracted, their topological properties do not change. To a topologist, there is no difference among a large square, a small disk, and a cone. But introduce a hole, and you have a different kind of topological object. The study of topology requires a solid background in calculus and a facility with logic and proofs.  A square, a disk, and a cone are topologically equivalent. A disk with a hole is different.

### Forgetting about Distance

Some have described topology as seeking the most general kind of space in which continuity can be defined. Remember that in calculus, a function is called continuous at a point a if  . And limit is defined using distance: f(x) must be "close to" f(a) whenever x is "close to" a. So how do you define continuity without using the concept of distance?

The answer lies in a simple idea from the real number line: x is close to, within ε of, a if and only if |x - a| < ε if and only if x is in the open interval (a-ε, a+ε). That is, saying the distance between two points is small is equivalent to saying that those two points are in a single open interval. The study of space is then liberated from distance by specifying which sets in the space are to be considered open. ### A Few Topological Questions

• How do you tell the difference between a solid disk and a disk with a hole in it?
• How do you tell the difference between a point on the interior of a disk and a point on the boundary?
• What distinguishes a line from a plane?
• What distinguishes a bowline knot from a clove hitch?
• What is the criterion for a space to be "connected"?
• How can a tiny human discover the shape of the earth she lives on? the shape of the universe?
• What properties must a space have so that sequences that look like they converge actually do?

### Topology as a Field

The first topological questions were asked centuries ago. Euler (1707-1783) was interested in a walking tour of the city of Königsberg. Möbius (1790-1868) discovered an interesting surface that has only one edge and one side. Poincaré (1854-1912) was interested in planetary motion and in the shape of the universe. These questions and others came together in the second half of the 19th century to comprise the field of topology. More on the history of topology can be found at the MacTutor History of Mathematics Archive.

Today, topology is a wide-spread field that is both theoretical and applied. The basic study begins with point-set topology, which comprises our introductory course. Other branches include geometric topology, algebraic topology, differential topology, and knot theory. Perhaps someone will invent "probabilistic topology" .... Recent applications are found in aerodynamics, chemistry, and computer networks. It may interest you to know that the solution sets to some differential equations are topological manifolds.