Topological space

In mathematics, a topological space is an ordered pair $(X,O)$ where $X$ is a set and $O$ is a certain collection of subsets of $X$ called the open sets or the topology of $X$. The topology of $X$ introduces a structure on the set $X$ which is useful for defining some important abstract notions such as the "closeness" of two elements of $X$ and convergence of sequences of elements of $X$.

Formal definition

A topological space is an ordered pair $(X,O)$ where $X$ is a set and $O$ is a collection of subsets of $X$ (i.e. $A\in O\Rightarrow A\subset X$) with the following three properties:

1. $X$ and $\emptyset$ (the empty set) are in $O$

2. The union of any number (countable or uncountable) of elements of $O$ is again in $O$

3. The intersection of any finite number of elements of $O$ is again in $O$

Elements of the set $O$ are called open sets (of $X$).

Note that as shorthand a topological space $(X,O)$ is often simply written as $X$ once the particular topology on $X$ is understood.

Examples

1. Let $X=\mathbb {R}$ where $\mathbb {R}$ denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:

${\mathopen {]}}a,b{\mathclose {[}}=\{y\in \mathbb {R} \mid a<y<b\}.$

Then a topology $O$ can be defined on $X=\mathbb {R}$ to consist of $\emptyset$ and all sets of the form:

$\bigcup _{\gamma \in \Gamma }{\mathopen {]}}a_{\gamma },b_{\gamma }{\mathclose {[}},$

where $\Gamma$ is any arbitrary index set, and $a_{\gamma }$ and $b_{\gamma }$ are real numbers satisfying $a_{\gamma }<b_{\gamma }$ for all $\gamma \in \Gamma$. This topology is precisely the familiar topology induced on $\mathbb {R}$ by the Euclidean distance $d(x,y)=|x-y|$ and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set $X$ and in the next example another topology on $\mathbb {R}$, albeit a relatively obscure one, will be constructed.

2. Let $X=\mathbb {R}$ as before. Let $O$ be a collection of subsets of $\mathbb {R}$ defined by the requirement that $A\in O$ if and only if $A=\emptyset$ or $A$ contains all except at most a finite number of real numbers. Then it is straightforward to verify that $O$ defined in this way has the three properties required to be a topology on $\mathbb {R}$. This topology is known as the Zariski topology.

Some topological notions

This section introduces some important topological notions. Throughout, X will denote a topological space with the topology O.

Partial list of topological notions

Neighbourhood

A subset N of X is a neighbourhood of a point $x\in X$ if N contains an open set $U\in O$ containing the point x

Limit point

A point $x\in X$ is a limit point of a subset A of X if any open set in O containing x also contains a point $y\in A$ with $y\neq x$. An equivalent definition is that $x\in X$ is a limit point of A if every neighbourhood of x contains a point $y\in A$ different from x.

Open cover

A collection ${\mathcal {U}}$ of open sets of X is said to be an open cover for X if each point $x\in X$ belongs to at least one of the open sets in ${\mathcal {U}}$

Path

A path $\gamma$ is a continuous function $\gamma :[0,1]\rightarrow X$. The point $\gamma (0)$ is said to be the starting point of $\gamma$ and $\gamma (1)$ is said to be the end point. A path joins its starting point to its end point

Hausdorff/separability property

X has the Hausdorff (or separability) property if for any pair $x,y\in X$ there exist disjoint sets U and V with $x\in U$ and $y\in V$

Connectedness

X is connected if given any two disjoint open sets U and V such that $X=U\cup V$, then either X=U or X=V

Path-connectedness

X is path-connected if for any pair $x,y\in X$ there exists a path joining x to y

Compactness

X is said to be compact if any open cover of X has a finite sub-cover. That is, any open cover has a finite number of elements which again constitute an open cover for X

A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.

Induced topologies

A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as induced topologies. Descriptions of some induced topologies are given below. Throughout, $(X,O_{X})$ will denote a topological space.

Some induced topologies

Relative topology

If A is a subset of X then open sets may be defined on A as sets of the form $O\cap A$ where O is any open set in $O_{X}$. The collection of all such open sets defines a topology on A called the relative topology of A as a subset of X

Quotient topology

If Y is another set and q is a surjective function from X to Y then open sets may be defined on Y as subsets U of Y such that $q^{-1}(U)=\{x\in X\mid q(x)\in U\}\in O_{X}$. The collection of all such open sets defines a topology on Y called the quotient topology induced by q

See also

Topology

Analysis

Metric space

References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980

2. D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [1]