= Metric realization of a countable star
In the <metric realization of a simplicial complex>, the point on arm $n$ at parameter $t$ is $(1-t)e_v+te_{u_n}$. Its distance from the common vertex is $\sqrt2\,t$. Consequently the vertex has a <neighbourhood basis> consisting of one uniformly short initial interval on every arm. Away from that vertex, each arm is locally isolated from the other arms.
The map $(n,t)\mapsto te^{i/n}$, with all $t=0$ identified, is a <homeomorphism> onto
$$
\{0\}\cup\{te^{i/n}:n\geq1,\ 0<t\leq1\}\subset\mathbb R^2.
$$
It is a bijection, preserves the one-dimensional topology on each arm away from zero, and matches the uniform-radius neighbourhood bases at zero. The positive real limiting ray is omitted. The outer endpoints form an infinite <closed discrete subset> in the original metric realization, so the image is not compact. This planar fan and the <weak realization of a countable star> have different topologies at their common vertex.
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