Minimal test sets for detecting discontinuities exist in the poset of convergent sequences and can be strictly smaller than the full family at points like those in the sequential fan.
Jech, Set Theory: The Third Millennium Edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin--Heidelberg
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On minimal collections of sequences for testing continuity
Minimal test sets for detecting discontinuities exist in the poset of convergent sequences and can be strictly smaller than the full family at points like those in the sequential fan.