In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.
A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighbourhood which is homeomorphic to real n-space Rn.
A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompactor second-countable.
In the remainder of this article a manifold will mean a topological manifold. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to Rn.
The property of being locally Euclidean is preserved by local homeomorphisms. That is, if X is locally Euclidean of dimension n and f : Y → X is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff spaces.
Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of σ-compactness and second-countability are the same. Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular.Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable. Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.
A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M, which are open sets since manifolds are locally-connected. Being locally path connected, a manifold is path-connected if and only if it is connected. It follows that the path-components are the same as the components.
The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need not be. It is true, however, that every locally Euclidean space is T1.
An example of a non-Hausdorff locally Euclidean space is the line with two origins. This space is created by replacing the origin of the real line with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This space is not Hausdorff because the two origins cannot be separated.
A manifold is metrizable if and only if it is paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold. In any case, non-paracompact manifolds are generally regarded as pathological. An example of a non-paracompact manifold is given by the long line. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces.
Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent.
Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.
Every compact manifold is second-countable and paracompact.
By invariance of domain, a non-empty n-manifold cannot be an m-manifold for n ≠ m.The dimension of a non-empty n-manifold is n. Being an n-manifold is a topological property, meaning that any topological space homeomorphic to an n-manifold is also an n-manifold.
By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of . Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domain that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in . Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:
A Euclidean neighborhood homeomorphic to an open ball in is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.
For any Euclidean neighborhood U, a homeomorphism is called a coordinate chart on U (although the word chart is frequently used to refer to the domain or range of such a map). A space M is locally Euclidean if and only if it can be covered by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M, together with their coordinate charts, is called an atlas on M. (The terminology comes from an analogy with cartography whereby a spherical globe can be described by an atlas of flat maps or charts).
Given two charts and with overlapping domains U and V, there is a transition function
Such a map is a homeomorphism between open subsets of . That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for differentiable manifolds the transition maps are required to be diffeomorphisms.
A 0-manifold is just a discrete space. A discrete space is second-countable if and only if it is countable.
Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle.
Every nonempty, compact, connected 2-manifold (or surface) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes.
A classification of 3-manifolds results from Thurston's geometrization conjecture, proven by Grigori Perelman in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
The full classification of n-manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory, which is known to be algorithmically undecidable.
In fact, there is no algorithm for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
A slightly more general concept is sometimes useful. A topological manifold with boundary is a Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space (for a fixed n):
Every topological manifold is a topological manifold with boundary, but not vice versa.
There are several methods of creating manifolds from other manifolds.
If M is an m-manifold and N is an n-manifold, the Cartesian product M×N is a (m+n)-manifold when given the product topology.
The disjoint union of a countable family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
The connected sum of two n-manifolds is defined by removing an open ball from each manifold and taking the quotient of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls. This results in another n-manifold.
Any open subset of an n-manifold is an n-manifold with the subspace topology.
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is . Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space, a much larger class of spaces.
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0, 1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
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