Topology
In topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation, an n-sphere is defined as a space homeomorphic In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of to the boundary of an (n+1)-ball; thus, it is homeomorphic In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of to the Euclidean n-sphere, but perhaps lacking its metric In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.
- a 0-sphere is a pair of points with the discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense
- a 1-sphere is a circle (up to In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent. In group theory, for example, homeomorphism In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of); thus, for example, (the image of) any knot In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as is a 1-sphere
- a 2-sphere is an ordinary sphere (up to In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent. In group theory, for example, homeomorphism In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of); thus, for example, any spheroid A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters is a 2-sphere
The n-sphere is denoted Sn. It is an example of a compact In mathematics, more specifically general topology and metric topology, an abstract mathematical space is said to be compact if, intuitively, one can take infinitely many "steps" in the space and still approach a point inside the space. Thus, while disks and spheres are compact, infinite lines and planes are not, nor is a disk or a topological manifold In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics without boundary In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. S is. A sphere need not be smooth In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and the surface of a ball are two-dimensional manifolds,; if it is smooth, it need not be diffeomorphic In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth to the Euclidean sphere.
The Heine-Borel theorem In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrary implies that a Euclidean n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. Sn is also bounded, therefore it is compact.
<<Table of Contents A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The | Next>> | Show All>>
