On minimal collections of sequences for testing continuity

TL;DR

This paper investigates the structure of minimal test sets of sequences for detecting discontinuities of real-valued functions at a point, revealing their existence, properties, and size advantages in specific spaces.

Contribution

It proves the existence of minimal elements in the poset of test sets under natural conditions and analyzes their structure and size in the sequential fan.

Findings

Existence of minimal test sets under natural hypotheses.

Some minimal test sets have a least element, others do not.

In the sequential fan, minimal test sets can be smaller than the full family of sequences.

Abstract

We study test sets: subfamilies of sequences converging to a point P that still suffice to detect every discontinuity of real-valued functions at P. Ordered by inclusion, these test sets form a poset. Under natural hypotheses at P, we prove that this poset has a minimal element. We also analyze its maximal chains, showing that some have a least element, while others do not. Finally, on the sequential fan we give a concrete realization in which the minimal test set produced by our construction has strictly smaller cardinality than the full family of convergent sequences.