A lens space is the 3-manifold obtained by gluing the boundaries of two solid tori together such that the meridian of the first goes to a -curve on the second, where a -curve wraps around the longitude times and around the meridian times.
According to most authors (e.g., Kelley 1955, p. 113; McCarty 1967, p. 144; Willard 1970, p. 92) a regular space is a topological space in which every neighborhood of a point contains a closed neighborhood of the same point.Another equivalent condition is the following: for every closed set and every point there are two disjoint open sets and such that and .In other sources (e.g., Bourbaki 1989, p. 80; Cullen 1968, p. 113) regularity is defined differently, using separation axioms.
A topological space that contains a homeomorphicimage of every topological space of a certain class.A metric space is said to be universal for a family of metric spaces if any space from is isometrically embeddable in . Fréchet (1910) proved that , the space of all bounded sequences of real numbers endowed with a supremum norm, is a universal space for the family of all separable metric spaces. Holsztynski (1978) proved that there exists a metric on , inducing the usual topology, such that every finite metric space embeds in (Ovchinnikov 2000).
The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in . Quotient spaces are also called factor spaces.This can be stated in terms of maps as follows: if denotes the map that sends each point to its equivalence class in , the topology on can be specified by prescribing that a subset of is open iff is open.In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compact metrizable space is a quotient of the Cantor set, any compact connected -dimensional manifold for is a quotient of any other, and a function out of a quotient space is continuous iff the function is continuous.Let be the closed -dimensional disk and its boundary, the -dimensional sphere. Then (which is homeomorphic to ), provides an example of..
Also called a -space, a Tychonoff space is a completely regular space (with the additional condition that it be for those authors who do not assume this in the definition of completely regular.) In any case, Tychonoff spaces can be characterized as the topological spaces which are homeomorphic to a subspace of some cube (equivalently, which are subspaces of some compact T2-space, or some T4-space.)
A projective space is a space that is invariant under the group of all general linear homogeneous transformation in the space concerned, but not under all the transformations of any group containing as a subgroup.A projective space is the space of one-dimensional vector subspaces of a given vector space. For real vector spaces, the notation or denotes the real projective space of dimension (i.e., the space of one-dimensional vector subspaces of ) and denotes the complex projective space of complex dimension (i.e., the space of one-dimensional complex vector subspaces of ). can also be viewed as the set consisting of together with its points at infinity.
A pointed space is a topological space together with a choice of a basepoint . The notation for a pointed space is . Maps between two pointed spaces must take basepoints to basepoints. Pointed spaces are widely used in algebraic topology, homotopy theory, and topological K-theory.
Teichmüller's theorem asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence relation. The equivalence classes form the Teichmüller space of compact Riemann surfaces of genus .
If is a normed linear space, then the set of continuous linear functionals on is called the dual (or conjugate) space of . When equipped with the norm , a dual normed space is a Banach space.
In continuum theory, a dendrite is a locally connected continuum that contains no simple closed curve. A semicircle is therefore a dendrite, while a triangle is not.The term dendrite is used by Steinhaus (1999, pp. 120-125) to refer to a system of line segments connecting a given set of points, where the total length of paths is as short as possible (therefore implying that no closed cycles are permitted) and the paths are not allowed to cross. This definition differs from the one in continuum theory since a semicircle is a dendritic continuum but is not a line segment.
A topological space having a countable dense subset. An example is the Euclidean space with the Euclidean topology, since it has the rational lattice as a countable dense subset and it is easy to show that every open -ball contains a point whose coordinates are all rational.
The set of all Schwartz functions is called a Schwartz space and is denoted . If denotes the set of smooth functions of compact support on , then this is a subset of . Since is dense in , is dense in for any .
Let be a locally Euclidean coordinate system. Then(1)Now plug in(2)(3)to obtain(4)Reading off the coefficients from(5)gives(6)(7)(8)Making a change of coordinates gives(9)(10)(11)(12)(13)(14)
A topological space, also called an abstract topological space, is a set together with a collection of open subsets that satisfies the four conditions: 1. The empty set is in . 2. is in . 3. The intersection of a finite number of sets in is also in . 4. The union of an arbitrary number of sets in is also in . Alternatively, may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become: 3. The intersection of an arbitrary number of sets in is also in . 4. The union of a finite number of sets in is also in . These axioms are designed so that the traditional definitions of open and closed intervals of the real line continue to be true. For example, the restriction in (3) can be seen to be necessary by considering , where an infinite intersection of open intervals is a closed set.In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff axioms (which..
Let be the set of continuous mappings . Then the topological space supplied with the compact-open topology is called a mapping space. If is a pointed space, then the mapping space of pointed maps is called the path space of . In words, is the space of all paths which begin at . is a contractible space with the contraction given by .
Let be a linear space over a field . Then the vector space tensor product is called a tensor space of degree . More specifically, a tensor space of type can be described as a vector space tensor product between copies of vector fields and copies of the dual vector fields, i.e., one-forms. For example,is the vector bundle of tensors on a manifold . Tensors of type form a vector space.
Euclidean -space, sometimes called Cartesian space or simply -space, is the space of all n-tuples of real numbers, (, , ..., ). Such -tuples are sometimes called points, although other nomenclature may be used (see below). The totality of -space is commonly denoted , although older literature uses the symbol (or actually, its non-doublestruck variant ; O'Neill 1966, p. 3). is a vector space and has Lebesgue covering dimension . For this reason, elements of are sometimes called -vectors. is the set of real numbers (i.e., the real line), and is called the Euclidean plane. In Euclidean space, covariant and contravariant quantities are equivalent so .
Two subspaces and of are said to be orthogonal if the dot product for all vectors and all .
Let be a point in an -dimensional compact manifold , and attach at a copy of tangential to . The resulting structure is called the tangent space of at and is denoted . If is a smooth curve passing through , then the derivative of at is a vector in .
According to many authors (e.g., Kelley 1955, p. 112; Joshi 1983, p. 162; Willard 1970, p. 99) a normal space is a topological space in which for any two disjoint closed sets there are two disjoint open sets and such that and .Other authors (e.g., Cullen 1968, p. 118) define the notion differently, using separation axioms.
In algebraic geometry classification problems, an algebraic variety (or other appropriate space in other parts of geometry) whose points correspond to the equivalence classes of the objects to be classified in some natural way. Moduli space can be thought of as the space of equivalence classes of complex structures on a fixed surface of genus , where two complex structures are deemed "the same" if they are equivalent by conformal mapping.
A topological space is compact if every open cover of has a finite subcover. In other words, if is the union of a family of open sets, there is a finite subfamily whose union is . A subset of a topological space is compact if it is compact as a topological space with the relative topology (i.e., every family of open sets of whose union contains has a finite subfamily whose union contains ).
Let be a real vector space (e.g., the real continuous functions on a closed interval , two-dimensional Euclidean space , the twice differentiable real functions on , etc.). Then is a real subspace of if is a subset of and, for every , and (the reals), and . Let be a homogeneous system of linear equations in , ..., . Then the subset of which consists of all solutions of the system is a subspace of .More generally, let be a field with , where is prime, and let denote the -dimensional vector space over . The number of -D linear subspaces of is(1)where this is the q-binomial coefficient(Aigner 1979, Exton 1983). The asymptotic limit is(2)where(3)(4)(5)(6)(Finch 2003), where is a Jacobi theta function and is a q-Pochhammer symbol. The case gives the q-analog of the Wallis formula...
The subset of the Euclidean plane formed by the union of the x-axis, the line segment with interval of the y-axis, and the sequence of segments with endpoints and for all positive integers .With respect to the relative topology is pathwise-connected. It is therefore connected, but not locally pathwise-connected at any point of the open interval since each open disk centered at point one of these points intersects in a union of parallel segments, forming a disconnected set.
The vector space generated by the columns of a matrix viewed as vectors. The column space of an matrix with real entries is a subspace generated by elements of , hence its dimension is at most . It is equal to the dimension of the row space of and is called the rank of .The matrix is associated with a linear transformation , defined byfor all vectors of , which we suppose written as column vectors. Note that is the product of an and an matrix, hence it is an matrix according to the rules of matrix multiplication. In this framework, the column vectors of are the vectors , where are the elements of the standard basis of . This shows that the column space of is the range of , and explains why the dimension of the latter is equal to the rank of .
The subset of the Euclidean plane formed by the union of the interval of the x-axis and all line segments of unit length passing through the origin which form an angle (measured in radians) with it, for all positive integers .With respect to the relative topology, is pathwise-connected. Therefore it is connected, but it is not locally pathwise-connected at any point of the open interval . Each disk centered at one of these points intersects in a union of disjoint segments, which form a disconnected set.Let be the broom space formed by segments of length for all natural numbers , and place , , , ... one right after the other on the -axis. This will cover the half-open interval of the -axis (above figure). The space obtained by adding the point (2,0) to this sequence of brooms is then connected im kleinen at point (2,0), since each open neighborhood of (2,0) contains a closed disk whose radius is exactly formed by the basis intervals of for all sufficiently..
A metric space is boundedly compact if all closed bounded subsets of are compact. Every boundedly compact metric space is complete. (This is a generalization of the Bolzano-Weierstrass theorem.)Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.
The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into the relevant properties of a given space.The everyday type of space familiar to most people is called Euclidean space. In Einstein's theory of Special Relativity, Euclidean three-space plus time (the "fourth dimension") are unified into the so-called Minkowski space. One of the most general type of mathematical spaces is the topological space.
Let be a finite dimensional vector space over the complex numbers, and let be a linear operator on . Then can be expressed as a direct sum of cyclic subspaces.
For , an open subset of , and , the Sobolev space is defined by(1)where , , and the derivatives are taken in a weak sense.When endowed with the norm(2) is a Banach space.In the special case , is denoted by . This space is a Hilbert space for the inner product(3)Sobolev spaces play an important role in the theory of partialdifferential equations.
A Banach space has the approximation property (AP) if, for every and each compact subset of , there is a finite rank operator in such that for each , .
The mathematical study of a nonlinear equation , where maps from a Hilbert space to a Hilbert space and which abstracts the construction of optical holograms.
The vector space generated by the rows of a matrix viewed as vectors. The row space of a matrix with real entries is a subspace generated by elements of , hence its dimension is at most equal to . It is equal to the dimension of the column space of (as will be shown below), and is called the rank of .The row vectors of are the coefficients of the unknowns in the linear equation system(1)where(2)and is the zero vector in . Hence, the solutions span the orthogonal complement to the row space in , and(3)On the other hand, the space of solutions also coincides with the kernel (or null space) of the linear transformation , defined by(4)for all vectors of . And it also true that(5)where denotes the kernel and the image, since the nullity and the rank always add up to the dimension of the domain. It follows that the dimension of the row space is(6)which is equal to the dimension of the column space...
Let be the set of continuous mappings . Then the topological space supplied with the compact-open topology is called a mapping space, and if is taken as the circle , then is called the "free loop space of " (or the space of closed paths).If is a pointed space, then a basepoint can be picked on the circle and the mapping space of pointed maps can be formed. This space is denoted and is called the "loop space of ."
Let be a connected topological space. Then is unicoherent provided that for any closed connected subsets and of , if , then is connected.An interval, say [0,1], is unicoherent, but a circle, say , is not unicoherent. An interesting example of a unicoherent space is a ray winding down on a circle. Specifically, let , where . Then the space , illustrated above, is unicoherent.
A vector space with a T2-space topology such that the operations of vector addition and scalar multiplication are continuous. The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a Hilbert space and a Banach space are topological vector spaces.The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the Banach space , one of the L-p-spaces, is used. But if one is interested in pointwise convergence, then no norm will suffice. Instead, for each define the seminormon the vector space of functions on . The seminorms define a topology, the smallest one in which the seminorms are continuous. So is equivalent to for all , i.e., pointwise convergence. In a similar way, it is possible to define a topology for which "convergence" means uniform convergence on compact sets...
Let be the set of all prime ideals of , and define . Then the Stone space of is the topological space defined on by postulating that the sets of the form are a subbase for the open sets.