 Weak realization of a countable star

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 $[0,{\epsilon }_n)$ on each arm, with no positive lower bound on the ${\epsilon }_n$.

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.