Topological space

Definition

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

• (T1) $X,\mathrm{\varnothing }\in \tau$,
• (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 $U:=U_1\cap \dots \cap U_n$ of open sets $U_1,\dots ,U_n\in \tau$ is an open set ($U\in \tau$)
• Let $X:=ℝ$ 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 $\bigcap _{n=1}^{\mathrm{\infty }}{U}_n$ is not open!.
• Let $X:=\{1,2,3,4,5\}$ and $T:=\{\{1,2,3\},\{2,3,4\},\{3,4,5\}\}$. 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