Fort space on a countably infinite set

Let $X=\omega\cup\{\infty\}=\{0,1,2\dots\}\cup\{\infty\}$. Define $U \subseteq X$ to be open if its complement either is finite or includes $\infty$.

This space is the one-point compactification of a countably infinite discrete space. It is homeomorphic to $\{0\}\cup\{2^{-n}:n<\omega\}$ as a subspace of Euclidean Real Numbers, and is also homeomorphic to the ordinal space $\omega+1$.

Defined as counterexample #23 ("Countable Fort Space") in DOI 10.1007/978-1-4612-6290-9.