# Topological space

## Definition

Consider ${\displaystyle X}$ to be a non-empty set, and also let ${\displaystyle \tau \subset \wp (X)}$ be a subset of the power set of ${\displaystyle X}$, such that an action ${\displaystyle \tau }$ fullfils the following conditions,

• ${\displaystyle X,\emptyset \in \tau }$,
• if ${\displaystyle U_{1},\ldots ,U_{n}\in \tau }$ then also the finite intersetion of these sets are element of the topology, i.e.
${\displaystyle U_{1}\cap \ldots \cap U_{n}\in \tau }$.
• let ${\displaystyle I}$ be an index set and for all ${\displaystyle i\in I}$ the subset ${\displaystyle U_{i}\subset X}$ is element of the topology (${\displaystyle U_{i}\in \tau }$) then also the union of these sets ${\displaystyle U_{i}}$ is an element of the topology <\math>, i.e.
${\displaystyle \bigcup _{i\in I}U_{i}\in \tau }$.

The pair ${\displaystyle (X,\tau )}$ is called topological space. Set sets in ${\displaystyle \tau \subset \wp (X)}$ are called the open sets in ${\displaystyle X}$.

• Let ${\displaystyle X:=\{1,2,3,4,5\}}$ and ${\displaystyle T:=\left\{\{1,2,3\},\{2,3,4\},\{3,4,5\}\right\}}$. Add a minimal number of sets, so ${\displaystyle T}$ and create ${\displaystyle \tau \supseteq T}$, so that ${\displaystyle (X,\tau )}$ is a topological space.