Concept
Topology
Overview
Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
that in a plane six equal circles may touch a seventh? that of all equal bodies, the sphere has the least surface? that [429] certain lines are incommensurable, and consequently ill-adapted for harmony? Do we not see that all these advantages or disadvantages spring from the idea of the thing, and that the contrary would imply contradiction? Can it be thought that the pain and discomfort of sentient creatures, and…
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
States what Topology is; the author is tied to Topology in the knowledge graph.
The relations between these things then receive "genaue und vollstaendige Beschreibung" through the axioms of the geometry. And the fact that these "things" are called points, lines, and planes is not to give to them any of the connotations ordinarily associated with these words further than are determined by the axiom groups that follow. Indeed, other geometers are even more explicit on this point.
John Dewey, Creative Intelligence
States what Topology is.
We may, if we please, call the proposition, “Three is two and one,” a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. But they are definitions in the geometrical sense, not the logical; asserting not the meaning of a term only, but along with it an observed matter of fact. The proposition, “A circle is a figure bounded by a line…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Topology is.
Philosophers who discussed Topology
- Gottfried Wilhelm Leibniz
- Marshall McLuhan
- Alfred Tarski
- Charles Sanders Peirce
- The Library of Babel
- Michel Serres
Related ideas
- Felix Hausdorff
- Greek language
- Georg Cantor
- Gottfried Wilhelm Leibniz
- Function (mathematics)
- Algebra
- Mathematics
- Geometry
- Molecule
- Real number
- Manifold
- Nobel Prize
- Quantum field theory
- Set (mathematics)
- Subset
- String theory
- Topological space
- Boolean algebra (structure)
- Nature (journal)
- Kazimierz Kuratowski
- Henri Poincaré
- Calculus
- Physical cosmology
- Category theory
Place in the tradition
Part of First-order logic.