Concepts of Topology
Concepts of Topology. Topological property. Topological transformation. Continuous deformation. Genus. Homeomorphism.
What is the subject of topology all about? With what subject matter is it concerned? What kinds of questions does it ask? With what kinds of problems is it concerned? Let us start with some dictionary definitions.
Topology. The study of those properties of geometric figures or solid bodies that remain invariant under certain transformations.
Funk & Wagnalls Dictionary
Topology. That branch of geometry which deals with the topological properties of figures. Combinatorial topology is the branch of topology which is the study of geometric forms by decomposing them into the simplest geometric figures (simplexes) which adjoin each other in a regular fashion. Algebraic topology includes the fields of topology which use algebraic methods (especially group theory) to a large extent. Point-set topology is the study of sets as accumulations of points (as contrasted to combinatorial methods of representing an object as a union of simpler objects) and describing sets in terms of topological properties such as being open, closed, compact, normal, regular, connected, etc.
James & James. Mathematics Dictionary.
Topological property. Any property of a geometrical figure A that holds as well for every figure into which A may be transformed by a topological transformation. Examples are the properties of connectedness and compactness, of subsets being open or closed, and of points being limit points. James & James. Mathematics Dictionary.
Def. Topological transformation. A transformation that carries a geometric figure A into another figure B is a topological transformation if the following conditions are met: 1) the
What is the subject of topology all about? With what subject matter is it concerned? What kinds of questions does it ask? With what kinds of problems is it concerned? Let us start with some dictionary definitions.
Topology. The study of those properties of geometric figures or solid bodies that remain invariant under certain transformations.
Funk & Wagnalls Dictionary
Topology. That branch of geometry which deals with the topological properties of figures. Combinatorial topology is the branch of topology which is the study of geometric forms by decomposing them into the simplest geometric figures (simplexes) which adjoin each other in a regular fashion. Algebraic topology includes the fields of topology which use algebraic methods (especially group theory) to a large extent. Point-set topology is the study of sets as accumulations of points (as contrasted to combinatorial methods of representing an object as a union of simpler objects) and describing sets in terms of topological properties such as being open, closed, compact, normal, regular, connected, etc.
James & James. Mathematics Dictionary.
Topological property. Any property of a geometrical figure A that holds as well for every figure into which A may be transformed by a topological transformation. Examples are the properties of connectedness and compactness, of subsets being open or closed, and of points being limit points. James & James. Mathematics Dictionary.
Def. Topological transformation. A transformation that carries a geometric figure A into another figure B is a topological transformation if the following conditions are met: 1) the