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Topological space

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.

Table of contents
1 Formal definition
2 Relations between topologies
3 Continuous functions
4 Alternative definitions
5 Examples of topological spaces
6 Constructing new topological spaces from given ones
7 Classification of topological spaces
8 Topological spaces with algebraic structure
9 History

Formal definition

Formally, a topological space is a set X together with a collection 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.

Relations between topologies

A variety of useful and not-so-useful topologies can be placed on nearly any set to form a topological space. When every set in a topology T1 is also found 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. Other terms, such as stronger, weaker, larger, and smaller are used in literature but with little agreement on their meanings.

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, Top, 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.)

The empty set and X are closed.
  • The intersection of any collection of closed sets is also closed.
  • The union of any pair of closed sets is also closed.

  • Examples of topological spaces

    Constructing new topological spaces from given ones

    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. Using these terms, we can give the following classification:

    Separation of points

    For a detailed treatment, see Separation axiom. Some of these terms are defined differently in older mathematical literature; see History of the separation axioms.


    Countability conditions



    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 topology.