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topology
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What is topology?
Topology is a field of mathematics, sometimes called "rubber sheet geometry," that studies the properties of geometric objects that remain unchanged through continuous deformations like bending, twisting, stretching, and shrinking, without tearing or gluing.
In topology, objects are considered equivalent if one can be continuously deformed into the other. Properties like being simply connected are of interest because they remain the same under continuous deformation.
The emergence of topology as a distinct field is associated with Henri Poincaré's 1895 publication, Analysis Situs, though topological ideas existed earlier. The field has two major subdisciplines: general topology, which deals with abstract objects, and algebraic topology, where algebraic objects are associated with topological spaces.
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How does topology differ from geometry?
Topology and geometry are distinct branches of mathematics. Geometry deals with shapes, spatial relationships, and space properties, originating from practical problems like surveying and is derived from Greek for "Earth measurement". In contrast, topology, also known as "rubber sheet geometry," considers objects equivalent if they can be continuously deformed into one another through bending, twisting, stretching, and shrinking, without tearing or gluing.
Geometry often involves numerically measured quantities such as lengths and angles, while topology focuses on qualitative similarities that remain unchanged through continuous deformations. Topology looks at objects in higher-dimensional spaces or abstract sets, requiring additional structure for discussing properties like function continuity.
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What is a topological space?
In topology, a topological space is a set X with a collection of subsets T that satisfies three axioms.
These include:
1. X and the empty set are members of T.
2. The intersection of any finite number of sets in T is in T.
3. The union of any collection of sets in T is in T.The sets in T are called open sets, and T is called a topology on X. The real number line becomes a topological space when its topology is specified as the collection of all possible unions of open intervals. A topological space may also be defined by an alternative set of axioms involving closed sets, which are complements of open sets. The generality of the axioms for a topological space permit mathematicians to view many sorts of mathematical structures, such as collections of functions in analysis, as topological spaces and thereby explain associated phenomena in new ways.
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What are common examples of shapes studied in topology?
Topology, also known as "rubber sheet geometry," studies properties that remain unchanged when objects are deformed through bending, twisting, stretching, or shrinking, without tearing or gluing.
Common examples of shapes studied in topology:
* Simple loops in a plane and the boundary edge of a square in a plane are topologically equivalent.
* The surface of a sphere is not topologically equivalent to a torus, which is the surface of a solid doughnut ring.
* Manifolds are topological spaces locally modeled on Euclidean space but with varying global properties.
* Knots are copies of a circle embedded in three-dimensional space.2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
How can objects be considered the same in topology even if they look different?
In topology, objects can be considered the same if one can be continuously deformed into the other without cutting or gluing. This is like imagining shapes made of rubber that you can stretch, bend, or twist, but not tear or join.
For example, a loop and a square are topologically equivalent because you can stretch a rubber band shaped like a loop to fit around a square. However, a sphere's surface and a donut's surface are not, because a small loop on a sphere can always shrink to a point, which isn't always possible on a donut. The math behind this involves one-to-one mapping that is continuous in both directions, called homeomorphism.
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What are some applications of topology in other areas of science or everyday life?
Topology, a branch of geometry, studies properties that remain unchanged through continuous deformations like stretching and folding. Applications of topology can be found in diverse fields.
Knot theory, a branch of algebraic topology, studies how knotted copies of a circle can be embedded in three-dimensional space and has potential applications in physics, chemistry, and biomedical engineering.
The winding and relaxation of supercoiled DNA is catalyzed by gyrases and topoisomerases. The supercoiling property is partially responsible for the great compaction of DNA that is necessary to fit it within the confines of the cell.
Differential topology and geometry have found applications in modern physics.
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News •
topology, branch of mathematics, sometimes referred to as “rubber sheet geometry,” in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while disallowing tearing apart or gluing together parts. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. Topology, while similar to geometry, differs from geometry in that geometrically equivalent objects often share numerically measured quantities, such as lengths or angles, while topologically equivalent objects resemble each other in a more qualitative sense.
The area of topology dealing with abstract objects is referred to as general, or point-set, topology. General topology overlaps with another important area of topology called algebraic topology. These areas of specialization form the two major subdisciplines of topology that developed during its relatively modern history.
Basic concepts of general topology
Simply connected
In some cases, the objects considered in topology are ordinary objects residing in three- (or lower-) dimensional space. For example, a simple loop in a plane and the boundary edge of a square in a plane are topologically equivalent, as may be observed by imagining the loop as a rubber band that can be stretched to fit tightly around the square. On the other hand, the surface of a sphere is not topologically equivalent to a torus, the surface of a solid doughnut ring. To see this, note that any small loop lying on a fixed sphere may be continuously shrunk, while being kept on the sphere, to any arbitrarily small diameter. An object possessing this property is said to be simply connected, and the property of being simply connected is indeed a property retained under a continuous deformation. However, some loops on a torus cannot be shrunk, as shown in the .
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Many results of topology involve objects as simple as those mentioned above. The importance of topology as a branch of mathematics, however, arises from its more general consideration of objects contained in higher-dimensional spaces or even abstract objects that are sets of elements of a very general nature. To facilitate this generalization, the notion of topological equivalence must be clarified.
Topological equivalence
The motions associated with a continuous deformation from one object to another occur in the context of some surrounding space, called the ambient space of the deformation. When a continuous deformation from one object to another can be performed in a particular ambient space, the two objects are said to be isotopic with respect to that space. For example, consider an object that consists of a circle and an isolated point inside the circle. Let a second object consist of a circle and an isolated point outside the circle, but in the same plane as the circle. In a two-dimensional ambient space these two objects cannot be continuously deformed into each other because it would require cutting the circles open to allow the isolated points to pass through. However, if three-dimensional space serves as the ambient space, a continuous deformation can be performed—simply lift the isolated point out of the plane and reinsert it on the other side of the circle to accomplish the task. Thus, these two objects are isotopic with respect to three-dimensional space, but they are not isotopic with respect to two-dimensional space.
The notion of objects being isotopic with respect to a larger ambient space provides a definition of extrinsic topological equivalence, in the sense that the space in which the objects are embedded plays a role. The example above motivates some interesting and entertaining extensions. One might imagine a pebble trapped inside a spherical shell. In three-dimensional space the pebble cannot be removed without cutting a hole through the shell, but by adding an abstract fourth dimension it can be removed without any such surgery. Similarly, a closed loop of rope that is tied as a trefoil, or overhand, knot (see ) in three-dimensional space can be untied in an abstract four-dimensional space.