Topology

When it comes to mathematics, topology (derived from the Greek words "o" meaning "place, location," and "study" meaning "study") is the study of the properties of a geometric object that are preserved under continuous deformations such as stretching, twisting, crumpling, and bending; that is, without the object being torn, glued, or passed through itself.

In mathematics, a topological space is a set that has been provided with a structure known as topology, which allows for the definition of continuous deformation of subspaces as well as more broadly, all types of continuity. In a topological space, any distance or metric determines its topology, as is the case with Euclidean spaces and, more broadly, metric spaces. Homomorphisms and homotopies are the types of deformations that are studied in topology. A topological property is a property that remains invariant in the face of such deformations. The dimension, which allows distinguishing in between line and a surface; compactness, which allows differentiating between a line and a circle; and connectedness, which allows distinguishing between a circle and two non-intersecting circles are all examples of topological properties that are fundamental to geometry.

Historically, the principles that underpin topology may be traced back to Gottfried Leibniz, who envisioned the geometria situs and the analysis situs in the seventeenth century. The Seven Bridges of Königsberg issue and the polyhedron formula, both developed by Leonhard Euler, are often considered to be the field's initial theorems. Topology was originally used in the nineteenth century by Johann Benedict Listing, but it was not until the early decades of the twentieth century that the concept of a topological space was conceptualised and formalised.