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. Define a ball $B_{r}(x)$ of radius $r>0$ around a point $x\in \mathbb {R}$ by:

$B_{r}(x)=\{y\in \mathbb {R} \mid |y-x|<r\}.$

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

$\cup _{\gamma \in \Gamma }B_{r_{\gamma }}(x_{\gamma }),$

where $\Gamma$ is any arbitrary index set, and $r_{\gamma }>0$ and $x_{\gamma }\in \mathbb {R}$ 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.

See also

Topology

Analysis