Metric realization of a countable star

In the metric realization of a simplicial complex, the point on arm $n$ at parameter $t$ is $(1-t)e_v+t{e}_{u_n}$. Its distance from the common vertex is $\sqrt{2}t$. 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 $(n,t)\mapsto t{e}^{i/n}$, with all $t=0$ identified, is a homeomorphism ontoIt 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.