Metric fan with $\omega$-many spines

The subspace of Euclidean Plane $\mathbb R^2$ defined by $X=\{\langle 0,0\rangle\}\cup\{\langle \frac{1}{m},\frac{1}{mn} \rangle :m,n\in\mathbb Z^+\}$

Equivalently, a classical special case of the filter fans defined in zbMATH 1339.54022: the set $X=\{\infty\}\cup\omega^2$ with $\omega^2$ discrete and neighborhoods of $\infty$ containing all but finitely-many rows of $\omega^2$.

Compare with the finer topology of Sequential fan with $\omega$-many spines.