one-point Lindelöfication of a discrete space of cardinality $\omega_1$ What is the nbhd of every point of one-point Lindelöfication of a discrete space of cardinality $\omega_1$? Could it be first countable?

A basic neighbourhood of a point $\alpha \in \omega_1$ (as a discrete set) is just $\\{\alpha\\}$ and a neighbourhood of $\omega_1$ (the added point) is of the form $\\{\omega_1\\} \cup (\omega_1 \setminus C)$ where $C$ is a countable subset of $\omega_1$.

This is not first countable at $\omega_1$: the countable intersection of countably many neighbourhoods of this point is again of the same form (as a countable union of countable sets is countable), and cannot be a singleton. This shows that at $\omega_1$ the space doesn't even have countable tighness.