Topological space

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 ${\displaystyle \tau }$ fullfils the following conditions,

• (T1) ${\displaystyle X,\emptyset \in \tau }$,
• (T2 - Intersection) if ${\displaystyle U_{1},U_{2}\in \tau }$ then also the finite intersetion of these sets are element of the topology, i.e.

${\displaystyle U_{1}\cap U_{2}\in \tau }$.

• (T2 - Union) 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, i.e.

${\displaystyle \bigcup _{i\in I}U_{i}\in \tau }$.

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

• Show that (T2) also implies, that any finite intersection ${\displaystyle U:=U_{1}\cap \ldots \cap U_{n}}$ of open sets ${\displaystyle U_{1},\ldots ,U_{n}\in \tau }$ is an open set (${\displaystyle U\in \tau }$)
• Let ${\displaystyle X:=\mathbb {R} }$ and ${\displaystyle \tau }$ be the standard euclidean topology generate by the absolute value ${\displaystyle |\cdot |}$. Provide a example of open sets ${\displaystyle U_{n}\in \tau }$ for which an infinite intersection ${\displaystyle \bigcap _{n=1}^{\infty }U_{n}}$ is not open!.
• 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.

• Topological vector space