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On uniform continuity of monotone functions
- Speaker(s)
- Piotr Zakrzewski
- Affiliation
- University of Warsaw
- Language of the talk
- English
- Date
- May 28, 2025, 4:15 p.m.
- Room
- room 5050
- Seminar
- Topology and Set Theory Seminar
The main subject of this talk is the statement that for every non-decreasing function f : [0, 1] −→ [0, 1], each subset of [0, 1] of cardinality c contains a set of cardinality c on which f is uniformly continuous.
Sierpiński proved that the statement is false under CH. On the other hand, it follows from the assumptions that d^∗ < c and c is regular, where 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. The first part of the talk will be devoted to a proof of this implication. In the second part I will sketch a proof that d^∗ = min{d, u}, where d is the dominating number and u is the ultrafilter number.
The results come from the joint paper with Roman Pol and Lyubomyr Zdomskyy "A note on uniform continuity of monotone functions", available at https://arxiv.org/abs/2502.20887
