= Finite stem of an infinite subset
{title2=$s\in[\mathbb N]^{<\omega}$}
= Finite stem
{synonym}
A <finite stem> is a finite initial segment $s$ of an infinite <subset> of $\mathbb N$. The associated <neighbourhood> in the <ordinary topology on infinite subsets> is $[s]=\{X:s\text{ is an initial segment of }X\}$. For an infinite tail $A$ above $\max s$, the <neighbourhood> $[s,A]=\{s\cup B:B\in[A]^\omega\}$ is basic in the <Ellentuck topology>. <Finite stems> allow a fusion argument to fix initial choices while successively thinning the remaining infinite tail.
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