Topological space

Definition

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

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

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

• (T2 - Union) let $I$ be an index set and for all $i\in I$ the subset $U_{i}\subset X$ is element of the topology ($U_{i}\in \tau$) then also the union of these sets $U_{i}$ is an element of the topology, i.e.

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

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

Learning Task

• Show that (T2) also implies, that any finite intersection $U:=U_{1}\cap \ldots \cap U_{n}$ of open sets $U_{1},\ldots ,U_{n}\in \tau$ is an open set ($U\in \tau$)
• Let $X:=\mathbb {R}$ and $\tau$ be the standard euclidean topology generate by the absolute value $|\cdot |$. Provide a example of open sets $U_{n}\in \tau$ for which an infinite intersection $\bigcap _{n=1}^{\infty }U_{n}$ is not open!.
• Let $X:=\{1,2,3,4,5\}$ and $T:=\left\{\{1,2,3\},\{2,3,4\},\{3,4,5\}\right\}$. Add a minimal number of sets, so $T$ and create $\tau \supseteq T$, so that $(X,\tau )$ is a topological space.

See also

• Topological vector space