Topology is the study of the mathematics of various shapes; it defines their features and the possible manipulations they can undergo while retaining their shape.
Topology is, in many ways, an umbrella term for various areas of mathematical theory which deal with concrete shapes and structures (though often doing so using strict mathematical formalism rather than models or pictures).
Encompassed under this umbrella are the theory of knots, the non-Euclidean geometry of general relativity, manifolds, symmetry groups, and ordinary, every day shapes. In essence, when discussing topology one is attempting to find simple ways to mathematically define certain shapes using a formal mathematical language.
Example: The Donut and the Coffee Cup
Surely the most overused example of a topological idea (that will not stop it from being echoed here) is the topological equivalence of a donut and a coffee cup. While it may seem counter-intuitive to say that these two objects are mathematically identical shapes, with a little thought, it should become clear.
In topology, one must begin by suspending disbelief and imagining that all objects in question are infinitely pliable. They can be bent, rotated, stretched or compacted by any degree. The only rule is that they may not be torn in any way (for an absolutely fascinating example of these rules, view this entertaining video on sphere transformations).
Now, presented with an infinitely pliable donut, one realizes that it is not difficult at all to transform it into a coffee cup. Simply create an indentation on one side, then continue to work this indentation until it is "bowl-shaped." Now, shrink the other half of the donut so that it is shaped like a normal coffee cup handle, and there it is.
This shape in topology is known as a "torus," and possesses many interesting mathematical features.
Homeomorphic Shapes
Any two shapes, such as the donut and the coffee cup, which can be stretched and morphed from one into the other without tearing is considered homeomorphic.
While it may seem at first like just about any shape, when sufficiently deformed, can be transformed into any other shape, this is not so. No matter how much a torus is bent, twisted or deformed, it can never be converted into a circle, or a donut with two holes.
Topological Examples
One of the first great examples of topology in history is Leonard Euler's 1736 paper, "Seven Bridges of Konigsberg," which explored a mathematical proof regarding whether or not it was possible to walk across each of this specific set of bridges exactly once.
This is a topological problem because it explores the use of space and the properties of a certain geometrical "shape" which could be deformed as much as necessary on paper while still retaining the same features of having seven bridges.
From here, there are a practically limitless variety of topological examples to choose from, as it has become an almost endlessly diverse field of study, relating to everything from logistics (practical applications derived from The Bridges of Konigsberg, perhaps), to physics (the curvature of space-time and string theory), to chemistry (the structure of DNA is topographical).
The subject of topology can also range from the simple (such as the donut/coffee cup example above, or any number of other rather simply explained examples) to the astonishingly complex (such as the mathematics of comparative topologies).
This branch of mathematics possesses numerous possibilities for mathematical advancements and surely promises any number of new surprises as it continues to be studied by the most intelligent of mathematicians.
See Also:
The Mobius Strip - an example of applied topology
References:
"Topology." Wolfram's Mathworld.
"The Colossal Book of Mathematics." Martin Gardner. W.W. Norton & Co. 2001.
