[Math Lair] Topology

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Topology is a branch of geometry that is sometimes described as the most general of all geometries. It is concerned with properties that are preserved under certain transformations referred to as continuous deformations. Continuous deformations can involve stretching, shrinking, or bending the object, but ripping or tearing is not permitted. Two figures are considered topologically identical, or homeomorphic, if one can be transformed into the other through continuous deformations. The traditional example is that a coffee cup is topologically equivalent to a doughnut (they both have one hole). Another example is that a cube would be homeomorphic to a sphere, and both would be homeomorphic to a tetrahedron.

[animation demonstrating topological equivalence of a coffee cup and a doughnut]

In topology, closed surfaces are classified according to their connectivity. The simplest form of surface is singly connected (genus 0), which can be illustrated by the surface of a sphere, a cube, or a lump of rock. A doubly connected surface (genus 1) is a torus. In general, the genus number can be considered to be the number of "holes" in a surface.