Chains in partially ordered spaces
Keywords:
partially ordered topological spaces, chains, total orders, semiclosed orders, order-dense, topological semilatticeAbstract
In this paper we study chains in partially ordered spaces, topological spaces equipped with a partial order, which we assume to be semiclosed, i.e., $\downarrow x$ and $\uparrow x$ are closed for each $x$. The strong order-theoretic property of being a chain allows us to develop some significant theory about chains in the weak setting of a semiclosed relation. We obtain conditions for chains to be compact, to be connected, and to have the relative topology agree with the order topology. We illustrate applications of these results in the setting of semitopological semilattices.
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