One-point compactification of the metric fan

Consider the set $\omega^2\cup\{\infty_x,\infty_y\}$ with points of $\omega^2$ isolated, neighborhoods of $\infty_x$ containing all but finitely-many columns, and neighborhoods of $\infty_y$ containing all but finitely-many rows.

Equivalently, the one-point compactification of Metric fan with $\omega$-many spines.