Calliope Ryan-Smith (Leeds)

Cardinal characteristics of the continuum

When speaking about subsets of the real number line, we may define some idea of smallness. When we do so, it is often the case that a countable union of small sets is not enough to cover the whole real line. Any list of countably many real numbers can be extended, so the shortest inextensible list of real numbers must be uncountable. Any countable collection of nowhere dense sets also does not cover the real line, so we need uncountably many of those to do that. The same holds for Lebesgue measure zero sets, strongly null sets, nowhere everywhere sets, and more. If aleph_0 isn't good enough then how many do we need?