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 $\{\}$ (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.

See also

Topology

Analysis