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Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in all branches of modern mathematics and can be seen as a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.

History

See topology.

Formal definition

Formally, a topological space is a set X together with a set T of subsets of X (i.e., T is a subset of the power set 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 called a topology on X. The sets in T are referred to as open sets, and their complements in X are called closed sets. The elements of X are often called points. Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.

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 bijective mapping 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 all topological spaces, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics. The attempt to classify the objects of this category 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 DeMorgan'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 function is continuous if the inverse image of every closed set is closed. The open sets can now be defined as the set of all complements of closed sets.

• The Kuratowski closure axioms determines 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 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 about 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

• The set of real numbers R is a topological space: the open sets are generated by the basis 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 R are topological spaces, and the open sets are generated by open balls.
• Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.
• The reals can also be given the upper-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 larger than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from below in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
• Every manifold is a topological space.
• Every simplex is a topological space. Simplexes are convex objects that are very useful in computational geometry. In 0, 1, 2 and 3 dimensional space 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 R or C 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.
• 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 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.
• 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.
• If Γ is an ordinal number, then the set is a topological space, generated by the intervals (a,b], where a and b are elements of Γ.

Constructing new topological spaces from given ones

• 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 f: X → Y is a function and X is a topological space, then Y gets a topology where a set is open if and only if its inverse image is open. A common example comes from an equivalence relation defined on the topological space X: the map f is then the natural projection on 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 according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets. A great many terms are used in topology to achieve these distinctions. These terms and definitions are collected together in the Topology Glossary.

Topological spaces with algebraic structure

It is almost universally true that all "large" algebraic objects carry a natural topology which is compatible with the algebraic operations. In order to study these objects, one typically has to take the topology into account. This leads to concepts such as topological groups, topological vector spaces and topological rings.

See also Algebraic topology.