Topological space

More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.

Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics.

relating the number of vertices (V), edges (E) and faces (F) of a convex polyhedron, and hence of a planar graph.

In 1827, Carl Friedrich Gauss published General investigations of curved surfaces, which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A.

"[3][non-primary source needed] Yet, "until Riemann's work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered".

The foundation of this science, for a space of any dimension, was created by Henri Poincaré.

[5] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.

Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory".

[6][7][better source needed] The utility of the concept of a topology is shown by the fact that there are several equivalent definitions of this mathematical structure.

The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of

A standard example of such a system of neighbourhoods is for the real line

The open sets then satisfy the axioms given below in the next definition of a topological space.

Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining

There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.

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

A proof that 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.

[11] This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function.

This example shows that in general topological spaces, limits of sequences need not be unique.

Similarly, every simplex and every simplicial complex inherits a natural topology from .

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 indexed family 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 infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

we construct a basis set consisting of all subsets of the union of the

The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space

is a variant of the Vietoris topology, and is named after mathematician James Fell.

consists of the so-called "marked metric graph structures" of volume 1 on

To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them.

Examples of such properties include connectedness, compactness, and various separation axioms.

Let be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set The bottom-left example is not a topology because the union of and [i.e. ] is missing; the bottom-right example is not a topology because the intersection of and [i.e. ], is missing.