Can the Hjalmar Ekdal topology be defined on uncountable sets? Can the Hjalmar Ekdal Topology be defined on uncountable sets and how would the various topological properties change from those associated with the set of positive integers?
(Example 55 in Counterexamples in Topology by Steen and Seebach)

Yes. The Hjalmar Ekdal space $H$ is homeomorphic to the product of the Sierpiński space and a countably infinite discrete space. Let $\kappa$ be any infinite cardinal, and let $D$ be the discrete space of cardinality $\kappa$. Let $S$ be the Sierpiński space; then $X=S\times D$ is the analogue of $H$ of cardinality $\kappa$, the points $\langle 0,d\rangle$ for $d\in D$ being analogous to the odd integers in $H$ as described by Steen & Seebach. $X$ has all of the properties that Steen & Seebach list for $H$, including the negative ones (e.g., $X$ is not compact).
