Let $[\omega]^\omega$ denote the collection of all infinite subsets of $\omega$. Let us call $S\subseteq [\omega]^\omega$ *lower-bounding* if for all $a\in [\omega]^\omega$ there is $s\in S$ such that $s\subseteq a$.


What is the minimum cardinality that a lower-bounding subset of $[\omega]^\omega$ can have?



Minimum cardinality of lower-bounding subset of $[\omega]^\omega$
They have size continuum, since there is an almost disjoint family of infinite sets of integers of size continuum.


