Discrete Topology on a Countably Infinite Set

Let $X=\omega=\{0,1,2,\dots\}$ be the space containing countably-many points and let all subsets of $X$ be open.

Defined as counterexample #2 ("Countable Discrete Topology") in DOI 10.1007/978-1-4612-6290-9.

This space is metrizable, with the standard metric for a discrete space given by $d(x,y)=1$ for $x\ne y$. A different metric that generates the same topology for $X$ is the Sierpinski metric ($d(x,y)=1+1/(x+y+2)$ for $x\ne y$ in $\omega$) given in counterexample #135 in DOI 10.1007/978-1-4612-6290-9. Since pi-base is meant to model topological properties and not properties of metrics, it only keeps track of topological spaces up to homeomorphism, and thus does not keep a separate entry for the Sierpinski metric.