Ramsey cone topology (${\tau }_{\mathrm{cone}}$)

This topology has basic open sets $[A{]}^{\omega }$ for infinite ground sets $A$, without fixed finite stems. It is coarser than the Ellentuck topology and differs from the ordinary topology on infinite subsets, whose basic cylinders fix an initial segment. The symbol $\tau$ is sometimes used for either coarser convention, so it should be accompanied by a definition.

The whole space of infinite subsets of the natural numbers is a meagre set in the Ramsey cone topology. For $D_j=\{X:\min X=j\}$, the closure is $\{X:j\in X\}$, which has empty interior: every infinite ground set can be thinned to omit $j$. Thus all $D_j$ are nowhere dense, while their countable union is the whole space. In contrast, the whole space is not meagre in either the Ellentuck topology or the ordinary topology on infinite subsets.