Topology

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.

Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together.

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once.

This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.

As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.

A famous example, known as the "Topologist's Breakfast", is that a topologist cannot distinguish a coffee mug from a doughnut; a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it while shrinking the hole into a handle.

[2] On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron.

[3] Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.

In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.

[4] The development of topology in the 20th century was marked by significant advances in both foundational theory and its application to other fields of mathematics.

[9] Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century.

The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".

The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions.

The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets.

The words nearby, arbitrarily small, and far apart can all be made precise by using open sets.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

[15] Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem.

In pointless topology one considers instead the lattice of open sets as the basic notion of the theory,[16] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories and with that the definition of general cohomology theories.

[17] Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects).

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA.

These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.

In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.

Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials.

[23] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.

It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics[26] by F.D.M Haldane.