Given a poset $(P,\leq)$ the *interval topology* on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.

Is $\mathcal{P}(\omega)/(fin)$ with the interval topology path-connected?

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