= Weak realization of a countable star
{title2=$W=\bigvee_{n\geq1}[0,1]_n$}
Identify the initial endpoints of countably many <closed intervals> to a vertex $v$, using the <quotient topology>. This is the <geometric realization> of the complex whose only edges are $\{v,u_n\}$. A <neighbourhood> of $v$ may contain a separately chosen initial interval $[0,\varepsilon_n)$ on each arm, with no positive lower bound on the $\varepsilon_n$.
The space is not <first countable>. Given any proposed countable <neighbourhood basis> $(U_n)$ at $v$, choose a point $x_n$ at coordinate $t_n>0$ on arm $n$ with $x_n\in U_n$. This is possible by the interval topology on that arm. The set whose intersection with arm $n$ is $[0,t_n/2)$ is an open neighbourhood $V$ of $v$, by the simplexwise definition. But $x_n\in U_n\setminus V$ for every $n$, so no $U_n$ lies inside $V$. Consequently this weak star cannot embed in the plane, or in any <metric space>.
The terminal vertices form an infinite <closed discrete subset>: on each closed arm they meet only its terminal endpoint. Hence the star is not <compact>, by <closed discrete subsets of compact spaces are finite>.
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2017-countable-star-topologies.png]
{title=Independent weak-star neighbourhoods and the metric fan topology}
{description=The finite schematic on the left illustrates different initial lengths on different arms. The planar fan on the right has uniform-radius neighbourhoods at the common vertex and carries the metric realization topology.}
Back to article page
