Topological space

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 collection 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 replaced by the equivalent one that the topology be closed under all finite intersections instead of just pairwise intersections. This has the benefit that we need not explicitly require that X be in T, since the empty intersection is (by convention) X. Similarly, we can conclude that the empty set is in T by using axiom 2 and taking a union over the empty collection. Nevertheless, it is conventional to include the first axiom even though it is redundant.

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 = {T_α : 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.) For example, 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.

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

An equivalent set of axioms is also the following:

1. The union of any number of sets belonging to T, also belongs to T.
2. The intersection of a finite number of sets belonging to T, belongs to Tau.

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.

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

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 set 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.

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 open intervals form a base or basis for the topology, meaning that every open set is a union of basic open sets. 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.

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 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.

Sierpinski space is the simplest non-trivial, non-discrete topological space. 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 any infinite set.

The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [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] may be endowed with the order topology 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, 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 X is a topological space and Y is a set, and if f : X  →  Y is a surjective function, then the quotient topology on Y is the set of inverse images of open. In other words, the quotient topology is the coursest topology on Y in which f is continuous. A common example of a quotient topology is 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, ..., U_n of open sets in X, we construct a basis set consisting of all subsets of the union of the U_i which have non-empty intersection with each U_i.

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

• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).

• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if and only if c({x}) ⊆ c({y}).

Specializations and generalizations

Topological spaces provide the most common notions of closeness and convergence for a space, but it may be possible in some cases to study more specialized or more general notions.

• Proximity spaces provide a notion of closeness of two sets.
• Metric spaces have a precise notion of distance between points, so that the closeness of any disparate pair of points can be compared.
• Uniform spaces carry a structure which axiomatize the idea of comparing the closeness of disparate pairs of points.
• Cauchy spaces carry a structure which axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.
• Convergence spaces carry a structure which captures some of the features of convergence of filters.
• σ-algebras provide a selection of sets whose size may be measured