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

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.

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]