Topological space - Wikipedia, the free encyclopedia Page 1 of 6 Topological space From Wikipedia, the free encyclopedia Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. Contents 1 Definition 1.1 Examples 1.2 Equivalent definitions 2 Comparison of topologies 3 Continuous functions 4 Examples of topological spaces Four examples and two non-examples of topologies on the 5 Topological constructions three-point set {1,2,3}. The bottom-left example is not a 6 Classification of topological topology because the union of {2} and {3} [i.e. {2,3}] is spaces missing; the bottom-right example is not a topology because 7 Topological spaces with algebraic the intersection of {1,2} and {2,3} [i.e. {2}], is missing. structure 8 Topological spaces with order structure 9 Specializations and generalizations 10 See also 11 References 12 External links Definition A topological space is a set X together with τ (a collection of subsets of X) satisfying the following axioms: 1. The empty set and X are in τ. 2. The union of any collection of sets in τ is also in τ. 3. The intersection of any pair of sets in τ is also in τ. The collection τ is called a topology on X. The elements of X are usually called points , though they can be any mathematical objects. A topological space in which the points are functions is called a function space. The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of X may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set . http://en.wikipedia.org/wiki/Topological_space 5/23/2011 Topological space - Wikipedia, the free encyclopedia Page 2 of 6 Examples 1. X = {1, 2, 3, 4} and collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms form a topology, the trivial topology. 2. X = {1, 2, 3, 4} and collection τ = {{}, {2}, {1,2}, {2,3}, {1,2,3}, {1,2,3,4}} of six subsets of X form another topology. 3. X = {1, 2, 3, 4} and collection τ = P(X) (the power set of X) form a third topology, the discrete topology. 4. X = Z, the set of integers, and collection τ equal to all finite subsets of the integers plus Z itself is not a topology, because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is not in τ. Equivalent definitions Main article: Characterizations of the category of topological spaces There are many other equivalent ways to define a topological space. (In other words, each of the following defines a category equivalent to the category of topological spaces above.) For example, using de Morgan's laws, the axioms defining open sets above become axioms defining closed sets: 1. The empty set and X are closed. 2. The intersection of any collection of closed sets is also closed. 3. The union of any pair of closed sets is also closed. Using these axioms, another way to define a topological space is as a set X together with a collection τ of subsets of X satisfying the following axioms: 1. The empty set and X are in τ. 2. The intersection of any collection of sets in τ is also in τ. 3. The union of any pair of sets in τ is also in τ. Under this definition, the sets in the topology τ are the closed sets, and their complements in X are the open sets. 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 X. A neighbourhood of a point x is any set that has an open subset containing x. The neighbourhood system at x consists of all neighbourhoods of x. A topology can be determined by a set of axioms concerning all neighbourhood systems. A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X the set of its accumulation points is specified. Comparison of topologies Main article: Comparison of topologies A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ τ τ τ τ τ 1 is also in a topology 2, we say that 2 is finer than 1, and 1 is coarser than 2. A proof that relies http://en.wikipedia.org/wiki/Topological_space 5/23/2011 Topological space - Wikipedia, the free encyclopedia Page 3 of 6 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 X forms a complete lattice: if F = { τα : α in A} is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X that contain every member of F. Continuous functions A function between topological spaces is called continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" 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. In category theory, Top , the category of topological spaces with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few. Examples of topological spaces 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. There are many ways of defining a topology on R, the set of real numbers. The standard topology on R 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 Rn can be given a topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C and Cn have a standard topology in which the basic open sets are open balls. 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. 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. http://en.wikipedia.org/wiki/Topological_space 5/23/2011 Topological space - Wikipedia, the free encyclopedia Page 4 of 6 Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations. A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. The Sierpi ński 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.