Topology
Jump to navigation
Jump to search
A Topology is a shape that is preserved under continuous deformations, such as stretching and bending (but not tearing or gluing).
- Context:
- It can range from being an Algebraic topology, to being a Differential Topology, to being a Geometric Topology.
- Example(s):
- Counter-Example(s):
- See: Location, Metric Space, Knot (Mathematics), Open Set, Topological Space, Connectedness, Compact (Topology).
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/topology Retrieved:2016-9-9.
- In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. [1] Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). 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. By the middle of the 20th century, topology had become a major branch of mathematics.
Topology has many subfields:
- General topology, also called point-set topology, establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It defines the basic notions used in all other branches of topology (including concepts like compactness and connectedness).
- Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
- Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
- Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.
- In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. [1] Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). 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. By the middle of the 20th century, topology had become a major branch of mathematics.
2008
- (Gertz, 2008) ⇒ Michael Gertz. (2008). “Lecture 2 - Spatial Concepts and Representation of Spatial Objects ECS 266 – Spatial Databases, UC Davis.
- Branch of mathematics that deals with spatial relationships that are preserved under bicontinuous deformation (stretching without tearing or gluing),
- sometimes called rubber-sheet geometry.
- The idea is that some geometric problems do not depend on the exact shape of objects involved, but rather on the way objects are “connected to each other”
- Spaces studied in topology are called topological spaces.
- Elementary topology is often called point-set topology