Fort Space on the Real Numbers

Let $X=\mathbb R\cup\{\infty\}$. Every point of $\mathbb R$ is isolated and the open neighborhoods of the point $\infty$ are the cofinite subsets of $X$ containing that point.

This space is the one-point compactification of an uncountable discrete space (compare with Fortissimo space on the real numbers and Fort space on a countably infinite set).

Defined as counterexample #24 ("Uncountable Fort Space") in DOI 10.1007/978-1-4612-6290-9.