Topology

Topology is the mathematical study of properties of objects which are preserved through
deformations, twistings, and stretchings. (Tearing, however, is not allowed.) A Circle is topologically equivalent to an Ellipse (into which it can be deformed by stretching) and a Sphere is equivalent to an Ellipsoid. Continuing along these lines, the Space of all positions of the minute hand on a clock is topologically equivalent to a Circle (where Space of all positions means ``the collection of all positions''). Similarly, the Space of all positions of the minute and hour hands is equivalent to a Torus. The Space of all positions of the hour, minute and second hands form a 4-D object that cannot be visualized quite as simply as the former objects since it cannot be placed in our 3-D world, although it can be visualized by other means.
 

There is more to topology, though. Topology began with the study of curves, surfaces, and other objects in the plane and 3-space. One of the central ideas in topology is that spatial objects like Circles and Spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are ``represented'' or ``embedded'' in space. For example, the statement ``if you remove a point from a Circle, you get a line segment'' applies just as well to the Circle as to an Ellipse, and even to tangled or knotted Circles, since the statement involves only topological properties.
 

Topology has to do with the study of spatial objects such as curves, surfaces, the space we call our universe, the space-time of general relativity, fractals, knots, manifolds (objects with some of the same basic spatial properties as our universe), phase spaces that are encountered in physics (such as the space of hand-positions of a clock), symmetry groups like the collection of ways of rotating a top,  etc.
 

The ``objects'' of topology are often formally defined as Topological Spaces. If two objects have the same topological properties, they are said to be Homeomorphic (although, strictly speaking, properties that are not destroyed by stretching and distorting an object are really properties preserved by Isotopy, not Homeomorphism; Isotopy has to do with distorting embedded objects, while Homeomorphism is intrinsic).
 

Topology is divided into Algebraic Topology (also called Combinatorial Topology), Differential
Topology, and Low-Dimensional Topology.
 

In Topology...
Isotopy has to do with distorting embedded objects, while Homeomorphism is intrinsic.

What is Topology ?

A simple definition of topology is that it is the study of properties that do NOT depend upon size or shape. The most fundamentalof topological properties is the number -- the number of parts, the number of intersections, the number of links, the number of holes, the number of dimensions..... Topological evolution is the study of when and how these numbers change.

Similarly, geometry may be said to be the study of properties that depend upon size and shape. It is extraordinary, but most current scientific and engineering concepts are based upon the geometric tradition, where the concept of number is constant, and the notion of topological evolution is ignored. Non-uniqueness and discontinuities are abhored. Yet there is evidence that irreversibility and aging imply changing topology.