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If we have any directed set $(D,\le)$ then we can add a point $\infty\notin D$ and then consider the topology on $C(D)=D\cup\{\infty\}$ such that all points of $D$ are isolated and the local base at $\infty$ consists of all "upper sets" $\langle d,\infty\rangle=\{\infty\}\cup\{d'\in D; d'\ge d\}$...