Topological space

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Topological spaces are structures that allow one to formalize 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.

This article is technical. For a general overview of the subject, see the article on topology.

Definition

A topological space is a set X together with a collection T of subsets of X satisfying the following axioms:

  1. The empty set and X are in T.
  2. The union of any collection of sets in T is also in T.
  3. The intersection of any pair of sets in T is also in T.

The set T is a topology on X. The sets in T are the open sets, and their complements in X are the closed sets. The elements of X are called points.

The requirement that the union of any collection of open sets be open is more stringent than simply requiring that all pairwise unions be open, as the former includes unions of infinite collections of sets. By induction, the intersection of any finite collection of open sets is open. Thus, the third axiom can be also formulated as: The intersection of any finite collection of sets in T is also in T.

It is conventional to specify explicitly, as we do above, that the empty set is in T. However, this is not in fact necessary, as the empty set is the union of the empty collection and must therefore be in T by rule 2.

Open sets and closed sets

An open set is any element in T as defined above. The complement of any set of T is , by definition, a closed set.

Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology T1 is also in a topology T2, we say that T2 is finer than T1, and T1 is coarser than T2. A proof which 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 X forms a complete lattice: if F = {Tα : 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 which contain every member of F.

Continuous functions

A function between topological spaces is said to be 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 said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

The category of topological spaces, Top, 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.

Alternative definitions

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.)

  • Using de Morgan's laws, the axioms defining open sets 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.
  • A neighbourhood of a point x is any set that contains an open set 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. Equivalently, a topology can be determined by a nearness relation between sets and points.
  • 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.

Examples of topological spaces

  • Any set can be given the discrete topology in which every set is open. The only convergent sequences or nets in this topology are those that are eventually constant.
  • Any set can be given the trivial 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.
  • The set of real numbers R is a topological space: the open sets are generated by the base of open intervals. This means a set is open if it is the union of (possibly infinitely many) open intervals. This is in many ways the most basic topological space and the one that guides most of our human intuition. However, relying on the real line as an intuitive guide for the general concept of topological space can often be dangerous.
  • More generally, the Euclidean spaces Rn are topological spaces, and the open sets are generated by open balls.
  • Every metric space is a topological space if one defines the open sets to be generated by the set of all open balls. In particular, every normed vector space is a topological space.
  • Many sets of 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 is a topological space.
  • Every simplex is a topological space. Simplexes are convex objects that are very useful in computational geometry. In Euclidean space of dimensions 0, 1, 2, and 3, the simplexes are the point, line segment, triangle and tetrahedron, respectively.
  • Every simplicial complex is a topological space. A simplicial complex is made up of many simplices. Many geometric objects can be modeled by simplicial complexes — see also Polytope.
  • The Zariski topology is a purely algebraically defined topology 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 is a topological space that generalises many of the geometric aspects of graphs with vertices and edges.
  • Sierpinski space is the simplest non-trivial, non-discrete topology. It has important relations to the theory of computation and semantics.
  • Any infinite 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 the set.
  • The real line can also be given the lower limit topology. Here, the open sets consist of the empty set, the whole real line, and all sets generated by half-open intervals of the form [a, b). This topology on R 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 Γ is an ordinal number, then the set [0, Γ] is a topological space, generated by the intervals (a, b], where a and b are elements of Γ.

Topological constructions

  • 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 nonempty collection of topological spaces, the product can be given the product topology. For finite products, the open sets are generated by the products of open sets.
  • A quotient space is defined as follows. If X is a topological space and Y is a set, and if f : X  →  Y is a surjective function, then Y gets a topology where a set is open if and only if its inverse image is open. A common example comes when an equivalence relation is defined on the topological space X: the map f 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 X is generated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consisting of all subsets of the union of the Ui which have non-empty intersection with each Ui.

Classification of topological spaces

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 which is not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms.

See the article on topological properties for more details and examples.

Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure which is not finite, we often have a natural topology which is 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 structure


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