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

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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.

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