Topological space

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]

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.

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 ${\displaystyle \{3\}}$ [i.e. ${\displaystyle \{2,3\}}$] is missing; the bottom-right example is not a topology because the intersection of ${\displaystyle \{1,2\}}$ and ${\displaystyle \{2,3\}}$ [i.e. ${\displaystyle \{2\}}$], 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 ${\displaystyle \tau .}$

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.

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.

Metric spaces embody a metric, a precise notion of distance between points.

Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

There are many ways of defining a topology on ${\displaystyle \mathbb {R} ,}$ the set of real numbers. The standard topology on ${\displaystyle \mathbb {R} }$ is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces ${\displaystyle \mathbb {R} ^{n}}$ can be given a topology. In the usual topology on ${\displaystyle \mathbb {R} ^{n}}$ the basic open sets are the open balls. Similarly, ${\displaystyle \mathbb {C} ,}$ the set of complex numbers, and ${\displaystyle \mathbb {C} ^{n}}$ have a standard topology in which the basic open sets are open balls.

Proximity spaces provide a notion of closeness of two sets.

Uniform spaces axiomatize ordering the distance between distinct points.

A topological space in which the points are functions is called a function space.

Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.

Convergence spaces capture some of the features of convergence of filters.

Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining sheaves.

If ${\displaystyle \Gamma }$ is a filter on a set ${\displaystyle X}$ then ${\displaystyle \{\varnothing \}\cup \Gamma }$ is a topology on ${\displaystyle X.}$

Many sets of linear 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 local field has a topology native to it, and this can be extended to vector spaces over that field.

Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from .

The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On ${\displaystyle \mathbb {R} ^{n}}$ or ${\displaystyle \mathbb {C} ^{n},}$ the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.

A linear graph has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges.

The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.

There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

Any 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 any infinite set.

Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals ${\displaystyle [a,b).}$ This topology on ${\displaystyle \mathbb {R} }$ is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.

If ${\displaystyle \Gamma }$ is an ordinal number, then the set ${\displaystyle \Gamma =[0,\Gamma )}$ may be endowed with the order topology generated by the intervals ${\displaystyle (a,b),}$ ${\displaystyle [0,b),}$ and ${\displaystyle (a,\Gamma )}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are elements of ${\displaystyle \Gamma .}$

Outer space of a free group ${\displaystyle F_{n}}$ consists of the so-called "marked metric graph structures" of volume 1 on ${\displaystyle F_{n}.}$[13]