In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. 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. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness.^[1][2] Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.

History

Around 1735, Leonhard Euler discovered the formula ${\displaystyle V-E+F=2}$ relating the number of vertices, edges and faces of a convex polyhedron, and hence of a planar graph. The study and generalization of this formula, specifically by Cauchy (1789-1857) and L'Huilier (1750-1840), boosted the study of topology. 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 infinitely small distance from A are deflected infinitely little from one and the same plane passing through A."^[3]

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".^[4] " Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not."^[4]

The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894.^[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". Metric spaces had been defined earlier in 1906 by Maurice Fréchet, though it was Hausdorff who popularised the term "metric space" (German: metrischer Raum).^[6][7]

Definitions

The utility of the concept of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatization suited for the application. The most commonly used is that in terms of open sets, but perhaps more intuitive is that in terms of neighbourhoods and so this is given first.

Definition via neighbourhoods

This axiomatization is due to Felix Hausdorff. Let ${\displaystyle X}$ be a set; the elements of ${\displaystyle X}$ are usually called points, though they can be any mathematical object. We allow ${\displaystyle X}$ to be empty. Let ${\displaystyle {\mathcal {N}}}$ be a function assigning to each ${\displaystyle x}$ (point) in ${\displaystyle X}$ a non-empty collection ${\displaystyle {\mathcal {N}}(x)}$ of subsets of ${\displaystyle X.}$ The elements of ${\displaystyle {\mathcal {N}}(x)}$ will be called neighbourhoods of ${\displaystyle x}$ with respect to ${\displaystyle {\mathcal {N}}}$ (or, simply, neighbourhoods of ${\displaystyle x}$). The function ${\displaystyle {\mathcal {N}}}$ is called a neighbourhood topology if the axioms below^[8] are satisfied; and then ${\displaystyle X}$ with ${\displaystyle {\mathcal {N}}}$ is called a topological space.

1. If ${\displaystyle N}$ is a neighbourhood of ${\displaystyle x}$ (i.e., ${\displaystyle N\in {\mathcal {N}}(x)}$), then ${\displaystyle x\in N.}$ In other words, each point belongs to every one of its neighbourhoods.
2. If ${\displaystyle N}$ is a subset of ${\displaystyle X}$ and includes a neighbourhood of ${\displaystyle x,}$ then ${\displaystyle N}$ is a neighbourhood of ${\displaystyle x.}$ I.e., every superset of a neighbourhood of a point ${\displaystyle x\in X}$ is again a neighbourhood of ${\displaystyle x.}$
3. The intersection of two neighbourhoods of ${\displaystyle x}$ is a neighbourhood of ${\displaystyle x.}$
4. Any neighbourhood ${\displaystyle N}$ of ${\displaystyle x}$ includes a neighbourhood ${\displaystyle M}$ of ${\displaystyle x}$ such that ${\displaystyle N}$ is a neighbourhood of each point of ${\displaystyle M.}$

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of ${\displaystyle X.}$

A standard example of such a system of neighbourhoods is for the real line ${\displaystyle \mathbb {R} ,}$ where a subset ${\displaystyle N}$ of ${\displaystyle \mathbb {R} }$ is defined to be a neighbourhood of a real number ${\displaystyle x}$ if it includes an open interval containing ${\displaystyle x.}$

Given such a structure, a subset ${\displaystyle U}$ of ${\displaystyle X}$ is defined to be open if ${\displaystyle U}$ is a neighbourhood of all points in ${\displaystyle U.}$ The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining ${\displaystyle N}$ to be a neighbourhood of ${\displaystyle x}$ if ${\displaystyle N}$ includes an open set ${\displaystyle U}$ such that ${\displaystyle x\in U.}$^[9]

Definition via open sets

A topology on a set X may be defined as a collection ${\displaystyle \tau }$ of subsets of X, called open sets and satisfying the following axioms:^[10]

1. The empty set and ${\displaystyle X}$ itself belong to ${\displaystyle \tau .}$
2. Any arbitrary (finite or infinite) union of members of ${\displaystyle \tau }$ belongs to ${\displaystyle \tau .}$
3. The intersection of any finite number of members of ${\displaystyle \tau }$ belongs to ${\displaystyle \tau .}$

As this definition of a topology is the most commonly used, the set ${\displaystyle \tau }$ of the open sets is commonly called a topology on ${\displaystyle X.}$

A subset ${\displaystyle C\subseteq X}$ is said to be closed in ${\displaystyle (X,\tau )}$ if its complement ${\displaystyle X\setminus C}$ is an open set.

Examples of topologies

Let ${\displaystyle \tau }$ be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set ${\displaystyle \{1,2,3\}.}$ The bottom-left example is not a topology because the union of ${\displaystyle \{2\}}$ and ${\displaystyle \{3\}}$ is missing; the bottom-right example is not a topology because the intersection of ${\displaystyle \{1,2\}}$ and ${\displaystyle \{2,3\}}$ , is missing.

1. Given ${\displaystyle X=\{1,2,3,4\},}$ the trivial or indiscrete topology on ${\displaystyle X}$ is the family ${\displaystyle \tau =\{\{\},\{1,2,3,4\}\}=\{\varnothing ,X\}}$ consisting of only the two subsets of ${\displaystyle X}$ required by the axioms forms a topology of ${\displaystyle X.}$
2. Given ${\displaystyle X=\{1,2,3,4\},}$ the family
   $${\displaystyle \tau =\{\{\},\{2\},\{1,2\},\{2,3\},\{1,2,3\},\{1,2,3,4\}\}=\{\varnothing ,\{2\},\{1,2\},\{2,3\},\{1,2,3\},X\}}$$

   
   of six subsets of ${\displaystyle X}$ forms another topology of ${\displaystyle X.}$
3. Given ${\displaystyle X=\{1,2,3,4\},}$ the discrete topology on ${\displaystyle X}$ is the power set of ${\displaystyle X,}$ which is the family ${\displaystyle \tau =\wp (X)}$ consisting of all possible subsets of ${\displaystyle X.}$ In this case the topological space ${\displaystyle (X,\tau )}$ is called a discrete space.
4. Given ${\displaystyle X=\mathbb {Z} ,}$ the set of integers, the family ${\displaystyle \tau }$ of all finite subsets of the integers plus ${\displaystyle \mathbb {Z} }$ itself is not a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of ${\displaystyle \mathbb {Z} ,}$ and so it cannot be in ${\displaystyle \tau .}$

Definition via closed sets

Using de Morgan's laws, the above axioms defining open sets become axioms defining closed sets:

1. The empty set and ${\displaystyle X}$ are closed.
2. The intersection of any collection of closed sets is also closed.
3. The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set ${\displaystyle X}$ together with a collection ${\displaystyle \tau }$ of closed subsets of ${\displaystyle X.}$ Thus the sets in the topology ${\displaystyle \tau }$ are the closed sets, and their complements in ${\displaystyle X}$ are the open sets.

Other definitions

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 ${\displaystyle X.}$

A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in ${\displaystyle X}$ the set of its accumulation points is specified.

Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology ${\displaystyle \tau _{1}}$ is also in a topology ${\displaystyle \tau _{2}}$ and ${\displaystyle \tau _{1}}$ is a subset of ${\displaystyle \tau _{2},}$ we say that ${\displaystyle \tau _{2}}$is finer than ${\displaystyle \tau _{1},}$ and ${\displaystyle \tau _{1}}$ is coarser than ${\displaystyle \tau _{2}.}$ 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.

The collection of all topologies on a given fixed set $$Zdroj:https://en.wikipedia.org?pojem=Topological_space
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