There is more to topology, though.
Prof. Ilya Kofman
Topology began with the study of curves, surfaces, and other objects in the plane and three-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 which are 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.
Topology can be used to abstract the inherent connectivity of objects while ignoring their detailed form. For example, the figures above illustrate the connectivity of a number of topologically distinct surfaces. In these figures, parallel edges drawn in solid join one another with the orientation indicated with arrows, so corners labeled with the same letter correspond to the same point, and dashed lines show edges that remain free Gardner , pp.
The labels are often omitted in such diagrams since they are implied by connection of parallel lines with the orientations indicated by the arrows.
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.
In particular, two mathematical objects are said to be homotopic if one can be continuously deformed into the other. Topology can be divided into algebraic topology which includes combinatorial topology , differential topology , and low-dimensional topology. The low-level language of topology, which is not really considered a separate "branch" of topology, is known as point-set topology.
There is also a formal definition for a topology defined in terms of set operations. A set along with a collection of subsets of it is said to be a topology if the subsets in obey the following properties:. The trivial subsets and the empty set are in. Whenever sets and are in , then so is.
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Whenever two or more sets are in , then so is their union. Bishop and Goldberg This definition can be used to enumerate the topologies on symbols. For example, the unique topology of order 1 is , while the four topologies of order 2 are , , , and. The numbers of topologies on sets of cardinalities , 2, OEIS A A set for which a topology has been specified is called a topological space Munkres , p. For example, the set together with the subsets comprises a topology, and is a topological space.
Topologies can be built up from topological bases. For the real numbers , a topological basis is the set of open intervals. Adamson, I. A General Topology Workbook. Alexandrov, P. Elementary Concepts of Topology. New York: Dover, Armstrong, M. Basic Topology, rev. New York: Springer-Verlag, Arnold, B. Intuitive Concepts in Elementary Topology. New York: Prentice-Hall, Barr, S.
Experiments in Topology. Berge, C. Bishop, R. Tensor Analysis on Manifolds. Blackett, D. New York: Academic Press, Bloch, E. Brown, J.
- Ancient Europe 8000 B.C.--A.D. 1000: encyclopedia of the Barbarian world;
- Home in Hollywood: The Imaginary Geography of Cinema.
- Basic Topology.
Chinn, W. Washington, DC: Math. Collins, G.
Comtet, L. Dordrecht, Netherlands: Reidel, p. Dugundji, J. Eppstein, D. Evans, J. ACM 10 , and , Francis, G. A Topological Picturebook. Gardner, M. New York: Scribner's, Gemignani, M. Elementary Topology. Gray, A. Each interaction is initiated by a specific agent--the client or Angel--and concluded by the other--the server or Demon.
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We present a category in which the objects--called interaction structures in the paper--serve as descriptions of services provided across such handshaken interfaces. The morphisms--called general simulations--model components that provide one such service, relying on another.
The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve in principle as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain. This category is then shown to coincide with the subcategory of "generated" basic topologies in Sambin's terminology, where a basic topology is given by a closure operator whose induced sup-lattice structure need not be distributive; and moreover, this operator is inductively generated from a basic cover relation.
This coincidence provides topologists with a natural source of examples for non-distributive formal topology. It raises a number of questions of interest both for formal topology and programming. The extra structure needed to make such a basic topology into a real formal topology is then interpreted in the context of interaction structures.