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What Is Topology?

By Ray Hawk
Updated May 21, 2024
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Topology is a branch of mathematics that deals with the study of surfaces or abstract spaces, where measurable quantities are not important. Due to this unique approach to mathematics, topology is sometimes referred to as rubber sheet geometry, because the shapes under consideration are imagined to exist on infinitely stretchable rubber sheets. In typical geometry, fundamental shapes such as the circle, square, and rectangle are the basis for all calculations, but, in topology, the basis is one of continuity and the position of points relative to one another.

A topological map can have points that together would make up a geometric shape such as a triangle. This collection of points is looked at as a space which remains unchanged; however, no matter how it is twisted or stretched, as the points on a rubber sheet, it would remain unchanged no matter in what form it was. This sort of conceptual framework for mathematics is often used in areas where large- or small-scale deformation often occurs, such as gravity wells in space, particle physics analysis at a sub-atomic level, and in the study of biological structures such as the changing shape of proteins.

The geometry of topology does not deal with the size of spaces, so a cube's surface area has the same topology as that of a sphere, as a person can imagine them being twisted to shift from one shape to the other. Such shapes that share identical features are referred to as homeomorphic. An example of two topological shapes that are not homeomorphic, or cannot be altered to resemble each other, are a sphere and a torus, or donut shape.

Discovering the core spatial properties of defined spaces is a primary goal in topology. A base level set topological map is referred to as a set of Euclidean spaces. Spaces are categorized by their number of dimensions, where a line is a space in one dimension, and a plane a space in two. The space that is experienced by human beings is referred to as three-dimensional Euclidean space. More complicated sets of spaces are called manifolds, which appear different on a local level than they do on a large scale.

Manifold sets and knot theory attempt to explain surfaces in many dimensions beyond what is perceivable on a literal human level, and the spaces are linked to algebraic invariants to classify them. This process of homotopy theory, or the relationship between identical topological spaces, was initiated by Henri Poincar&eacute, a French mathematician who lived from 1854 to 1912. Mathematicians have proven Poincar&eacute's work in all dimensions but three, where complete classification schemes for topologies remains elusive.

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