Topology studies surfaces and abstract spaces without focusing on measurable quantities. Shapes are imagined on infinitely stretchable rubber sheets, and continuity and position of points are the basis for calculations. Topological maps remain unchanged despite stretching or twisting. Spaces are classified by dimensions, and manifolds appear different locally than at large scales. Multiple sets and knot theory explain surfaces in many dimensions, and spaces are connected to algebraic invariants to classify them. Henri Poincaré started the theory of homotopy, but complete classification schemes for topologies in three dimensions remain elusive.
Topology is a branch of mathematics that deals with the study of surfaces or abstract spaces, where measurable quantities are not important. Because of this unique approach to mathematics, topology is sometimes referred to as rubbersheet geometry, because the shapes under study are imagined to exist on infinitely stretchable sheets of rubber. In typical geometry, fundamental shapes such as the circle, square and rectangle are the basis for all calculations, but, in topology, the basis is that of the continuity and position of points with respect to each other.
A topological map can have points that together would form a geometric shape such as a triangle. This collection of points is seen as a space that remains unchanged; however, no matter how twisted or stretched, like the dots on a rubber sheet, it would remain unchanged no matter what shape it was in. This type of conceptual framework for mathematics is often used in areas where large- or small-scale deformations often occur, such as gravity wells in space, the analysis of particle physics at the subatomic level, and in the study of biological structures such as the change in the shape of proteins.
The geometry of topology is not concerned with the dimension of spaces, so the surface area of a cube has the same topology as that of a sphere, as a person can imagine it twisted to go from one shape to another. Such forms that share identical characteristics are referred to as homeomorphic. An example of two topological shapes that are not homeomorphic, or that cannot be altered to look alike, are a sphere and a torus, or a donut shape.
Discovering the fundamental spatial properties of defined spaces is a primary goal in topology. A topological map of a base-level set is called a set of Euclidean spaces. Spaces are classified according to their number of dimensions, where a line is a space in one dimension and a plane is 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 locally than they do at large scales.
Multiple sets and knot theory attempt to explain surfaces in many dimensions beyond what is literally humanly perceivable, and spaces are connected to algebraic invariants to classify them. This process of theory of homotopy, or the relationship between identical topological spaces, was started by Henri Poincaré, a French mathematician who lived from 1854 to 1912. Mathematicians have proved Poincaré’s work in all dimensions except three, in which the complete classification schemes for topologies remain elusive.
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