Acceptance and rejection of finite stems
= Acceptance and rejection of finite stems
Relative to a family $Y$, an infinite reservoir accepts a finite stem $s$ when its entire <Ellentuck topology> neighbourhood $[s,A]$ lies in $Y$. It rejects $s$ when no infinite refinement accepts it. Every reservoir has a refinement deciding a prescribed stem, and both outcomes are hereditary under further infinite thinning. These definitions turn open-set homogeneity into a fusion construction.