A base of a funcoid which is not a filter base

The converse of this theorem does not hold.

Counterexample: Take S = \mathrm{up}\, \mathrm{id}^{\mathsf{FCD}}_{\Omega}. We know that S is not a filter base. But it is trivial to prove that S is a base of the funcoid \mathrm{id}^{\mathsf{FCD}}_{\Omega}.

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