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.
Hru s \' a k, Almost disjoint families and topology, in Recent Progress in General Topology III (K
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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.