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Topological spaces are structures which 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 which 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:

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

The set T is also called a topology on X. The sets in T are referred to as open sets, and their complements in X are called closed sets. Roughly speaking, open sets are thought of as neighborhoods of points; two points are "close together" if there are many open sets that contain both of them.

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 points which are "close together" get mapped to points which are "close together".

Alternative definitions

There are many other equivalent ways to define a topological space. Instead of defining open sets, it is possible to define first the closed sets, with the properties that the intersection of arbitrarily many closed sets is closed, the union of a finite number of closed sets is closed, and X and the empty set are closed. Open sets are then defined as the complements of closed sets. Another method is to define the topology by means of the closure operator. The closure operator is a function from the power set of X to itself which satisfies the following axioms (called the Kuratowski closure axioms): the closure operator is idempotent, every set is a subset of its closure, the closure of the empty set is empty, and the closure of the union of two sets is the union of their closures. Closed sets are then the fixed points of this operator. A topology on X is also completely determined if for every net in X the set of its limits is specified.

Examples of topological spaces

• The real numbers R: the open sets are unions 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.
• More generally, every interval in R is a topological space, and so are the Euclidean spaces R.
• The complex numbers C: the open sets are unions of open discs.
• Any metric space turns into a topological space if we define a set to be open if it is a (possibly infinite) union of open balls. This includes useful infinite dimensional spaces like the Banach spaces and Hilbert spaces studied in functional analysis.
• Manifolds. In particular, surfaces.
• A simplex. 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.
• Simplicial complexes. A simplicial complex is made up of many simplices. Many geometric objects can be modeled by simplicial complexes -- see also Polytope.
• CW complexes. Glued together cells that inherit essentially the quotient topology.
• The Zariski topology. 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.
• The weak topology. A useful topology for operators studied in functional analysis.

• Any set with the discrete topology (i.e., every set is open, which has the effect that no two points are "close" to each other).
• Any set with the trivial topology (i.e., only the empty set and the whole space are open, which has the effect of "lumping all points together").
• Any infinite set with the cofinite topology (i.e., the open sets are the empty set and the sets whose complement is finite). This is the smallest T₁ topology on the set.

Constructing new topological spaces from given ones

• A subset of a topological space. The open sets are the intersections of the open sets of the larger space with the subset. This is also called a subspace.
• Products of topological spaces. For finite products, the open sets are the sets that are unions of products of open sets.
• Quotient spaces. 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. 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.