**Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.**

*General topology*, or*point-set topology*, defines and studies properties of spaces and maps such as**connectedness**

**,**

**compactness**

**and**

**continuity**

**.**

*Algebraic topology*uses structures from**abstract algebra**

**, especially the**

**group**

**to study topological spaces and the maps between them.**

**The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.**

**One of the first papers in topology was the demonstration, by**

**Leonhard Euler**

**, that it was impossible to find a route through the town of Königsberg (now**

**Kaliningrad**

**) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the**

*Seven Bridges of Königsberg*, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as**graph theory**

**.**

**Similarly, the**

**hairy ball theorem**

**of algebraic topology says that "one cannot comb the hair on a ball smooth." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing**

**continuous**

**tangent vector**

**field**

**on the**

**sphere**

**. As with the**

*Bridges of Königsberg*, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of blob (subject to certain conditions on the smoothness of the surface), as long as it has no holes.**In order to deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems**

*do*rely on. From this need arises the notion of*topological equivalence*. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere.**Intuitively, two spaces are topologically equivalent if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't tell the**

**coffee mug**

**out of which she is drinking from the**

**doughnut**

**she is eating, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.**

**A simple introductory exercise is to classify the lowercase letters of the**

**English alphabet**

**according to topological equivalence. (The lines of the letters are assumed to have non-zero width.) In most fonts in modern use, there is a class {a, b, d, e, o, p, q} of letters with one hole, a class {c, f, h, k, l, m, n, r, s, t, u, v, w, x, y, z} of letters without a hole, and a class {i, j} of letters consisting of two pieces. g may either belong in the class with one hole, or (in some fonts) it may be the sole element of a class of letters with two holes, depending on whether or not the tail is closed. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several different classifications depending on which font is used. Letter topology is of practical relevance in stencil typography: The font**

**Braggadocio**

**, for instance, can be cut out of a plane without falling apart.**

**A continuous deformation (**

**homotopy**

**) of a coffee cup into a doughnut (**

**torus**

**) and back.**