= Cone on an infinite coinfinite ground set
{title2=$E_A=[A]^\omega$}
For an infinite <coinfinite set> $A\subseteq\mathbb N$, let $E_A=[A]^\omega$ in the <space of infinite subsets of the natural numbers>. This cone is a nonempty <clopen set> in the <Ellentuck topology>, because it is the basic <neighbourhood> $[\varnothing,A]$, and any $X\notin E_A$ has a finite initial segment containing a point outside $A$ and hence a basic <neighbourhood> disjoint from $E_A$.
The cone is a <closed set> with empty <interior> in the <ordinary topology on infinite subsets>: membership of a point outside $A$ witnesses its complement as an <open set>, while every ordinary finite-stem <neighbourhood> can be extended by an arbitrarily large point outside $A$. Thus $E_A$ is an ordinary <nowhere dense set>, but it is neither ordinary open nor Ellentuck nowhere dense. It provides both distinctions between these two <topologies>.
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