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 X of A if any open set in O containing x also contains a point $y\in A$ with $y\notin x$ . 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 opens 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