Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness.[1] Other spaces, such as Euclidean spaces, metric spaces and manifolds, are topological spaces with extra structures, properties or constraints.

Although very general, topological spaces are a fundamental concept used in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

HistoryEdit

Around 1735, Leonhard Euler discovered the formula   relating the number of vertices, edges and faces of a convex polyhedron, and hence of a planar graph. The study and generalization of this formula, specifically by Cauchy and L'Huilier, is at the origin of topology. In 1827, Carl Friedrich Gauss published General investigations of curved surfaces which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A."[2]

Yet, "until Riemann’s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered."[3] "Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not."[3]

The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894.[4] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.

Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet, though it was Hausdorff who introduced the term "metric space".[citation needed]

DefinitionsEdit

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatization suited for the application. The most commonly used is that in terms of open sets, but perhaps more intuitive is that in terms of neighbourhoods and so this is given first.

Definition via neighbourhoodsEdit

This axiomatization is due to Felix Hausdorff. Let   be a set; the elements of   are usually called points, though they can be any mathematical object. We allow   to be empty. Let   be a function assigning to each   (point) in   a non-empty collection   of subsets of   The elements of   will be called neighbourhoods of   with respect to   (or, simply, neighbourhoods of  ). The function   is called a neighbourhood topology if the axioms below[5] are satisfied; and then   with   is called a topological space.

  1. If   is a neighbourhood of   (i.e.,  ), then   In other words, each point belongs to every one of its neighbourhoods.
  2. If   is a subset of   and includes a neighbourhood of   then   is a neighbourhood of   I.e., every superset of a neighbourhood of a point   is again a neighbourhood of  
  3. The intersection of two neighbourhoods of   is a neighbourhood of  
  4. Any neighbourhood   of   includes a neighbourhood   of   such that   is a neighbourhood of each point of  

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of  

A standard example of such a system of neighbourhoods is for the real line   where a subset   of   is defined to be a neighbourhood of a real number   if it includes an open interval containing  

Given such a structure, a subset   of   is defined to be open if   is a neighbourhood of all points in   The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining   to be a neighbourhood of   if   includes an open set   such that  [6]

Definition via open sets Edit

 
Four examples and two non-examples of topologies on the three-point set   The bottom-left example is not a topology because the union of   and   [i.e.  ] is missing; the bottom-right example is not a topology because the intersection of   and   [i.e.  ], is missing.

A topological space is an ordered pair   where   is a set and   is a collection of subsets of   satisfying the following axioms:[7]

  1. The empty set and   itself belong to  
  2. Any arbitrary (finite or infinite) union of members of   belongs to  
  3. The intersection of any finite number of members of   belongs to  

The elements of   are called open sets and the collection   is called a topology on   A subset   is said to be closed in   if and only if its complement   is an element of  

Examples of topologiesEdit

  1. Given   the trivial or indiscrete topology on   is the family   consisting of only the two subsets of   required by the axioms forms a topology of  
  2. Given   the family
     
    of six subsets of   forms another topology of  
  3. Given   the discrete topology on   is the power set of   which is the family   consisting of all possible subsets of   In this case the topological space   is called a discrete space.
  4. Given   the set of integers, the family   of all finite subsets of the integers plus   itself is not a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of   and so it cannot be in  

Definition via closed setsEdit

Using de Morgan's laws, the above axioms defining open sets become axioms defining closed sets:

  1. The empty set and   are closed.
  2. The intersection of any collection of closed sets is also closed.
  3. The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set   together with a collection   of closed subsets of   Thus the sets in the topology   are the closed sets, and their complements in   are the open sets.

Other definitionsEdit

There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.

Another way to define a topological space is by using the Kuratowski closure axioms, which define the closed sets as the fixed points of an operator on the power set of  

A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in   the set of its accumulation points is specified.

Comparison of topologiesEdit

A variety of topologies can be placed on a set to form a topological space. When every set in a topology   is also in a topology   and   is a subset of   we say that  is finer than   and   is coarser than   A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.

The collection of all topologies on a given fixed set   forms a complete lattice: if   is a collection of topologies on   then the meet of   is the intersection of   and the join of   is the meet of the collection of all topologies on   that contain every member of  

Continuous functionsEdit

A function   between topological spaces is called continuous if for every and every neighbourhood   of   there is a neighbourhood   of   such that   This relates easily to the usual definition in analysis. Equivalently,   is continuous if the inverse image of every open set is open.[8] This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.[9]

In category theory, one of the fundamental categories is Top, which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory.

Examples of topological spacesEdit

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique.

Metric spacesEdit

Metric spaces embody a metric, a precise notion of distance between points.

Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

There are many ways of defining a topology on   the set of real numbers. The standard topology on   is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces   can be given a topology. In the usual topology on   the basic open sets are the open balls. Similarly,   the set of complex numbers, and   have a standard topology in which the basic open sets are open balls.

Proximity spacesEdit

Proximity spaces provide a notion of closeness of two sets.

Uniform spacesEdit

Uniform spaces axiomatize ordering the distance between distinct points.

Function spacesEdit

A topological space in which the points are functions is called a function space.

Cauchy spacesEdit

Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.

Convergence spacesEdit

Convergence spaces capture some of the features of convergence of filters.

Grothendieck sitesEdit

Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining sheaves.

Other spacesEdit

If   is a filter on a set   then   is a topology on  

Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

Any local field has a topology native to it, and this can be extended to vector spaces over that field.

Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from .

The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On   or   the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.

A linear graph has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges.

The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.

There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set.[citation needed]

Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals   This topology on   is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.

If   is an ordinal number, then the set   may be endowed with the order topology generated by the intervals     and   where   and   are elements of  

Outer space of a free group   consists of the so-called "marked metric graph structures" of volume 1 on  [10]

Topological constructionsEdit

Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

A quotient space is defined as follows: if   is a topological space and   is a set, and if   is a surjective function, then the quotient topology on   is the collection of subsets of   that have open inverse images under   In other words, the quotient topology is the finest topology on   for which   is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space   The map   is then the natural projection onto the set of equivalence classes.

The Vietoris topology on the set of all non-empty subsets of a topological space   named for Leopold Vietoris, is generated by the following basis: for every  -tuple   of open sets in   we construct a basis set consisting of all subsets of the union of the   that have non-empty intersections with each  

The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space   is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every  -tuple   of open sets in   and for every compact set   the set of all subsets of   that are disjoint from   and have nonempty intersections with each   is a member of the basis.

Classification of topological spacesEdit

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms. For algebraic invariants see algebraic topology.

Topological spaces with algebraic structureEdit

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields.

Topological spaces with order structureEdit

See alsoEdit

CitationsEdit

  1. ^ Schubert 1968, p. 13
  2. ^ Gauss 1827.
  3. ^ a b Gallier & Xu 2013.
  4. ^ J. Stillwell, Mathematics and its history
  5. ^ Brown 2006, section 2.1.
  6. ^ Brown 2006, section 2.2.
  7. ^ Armstrong 1983, definition 2.1.
  8. ^ Armstrong 1983, theorem 2.6.
  9. ^ Munkres, James R (2015). Topology. pp. 317–319. ISBN 978-93-325-4953-1.
  10. ^ Culler, Marc; Vogtmann, Karen (1986). "Moduli of graphs and automorphisms of free groups" (PDF). Inventiones Mathematicae. 84 (1): 91–119. doi:10.1007/BF01388734.

BibliographyEdit

External linksEdit