Abstract:
A common observation throughout the 20th century made in a variety of contexts
in real analysis and topology is that “countable” implies “small”. Examples of this
include the Baire category theorem (countable unions of closed nowhere dense
sets of reals cannot cover the real line), and the fact that countable sets have
Lebesgue measure zero. Set theorists have devoted much time to trying to
understand when “countable” can be replaced by sets of larger cardinality. These
investigations have betrayed a deep connection between topology and analysis of
the real line on one hand and combinatorics of infinite sets of natural numbers on
the other. In this talk I will survey some of the background on these topics before
turning to my own research on these subjects. I will show in particular how classical
combinatorial objects such as trees, linear orders and various special families of
infinite sets of real numbers can give surprising information about the topological
and analytic properties of the real line.