2025/02/28 by Pol, Roman, Zakrzewski, Piotr, Zdomskyy, Lyubomyr
#03E17 #03E20 #26A15 #54C05 #54E45 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · doi:10.48550/arxiv.2502.20887
We prove that it is consistent with ZFC that for every non-decreasing function f:[0,1]→ [0,1], each subset of [0,1] of cardinality \mathfrak c contains a set of cardinality \mathfrak c on which f is uniformly continuous. We show that this statement follows from the assumptions that \mathfrak d^* < \mathfrak c and \mathfrak c is regular, where \mathfrak d^*≤ \mathfrak d is the smallest cardinality κ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most κ-many open sets in the Cantor set. We establish also that \mathfrak d^*=min\\mathfrak u, \mathfrak d\=min\\mathfrak r, \mathfrak d\, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.