# Structures closed under intersection and minimality

Mathematical structures that are closed under countable or uncountable intersections often give rise to a minimal structure generated by some set. We would go through examples in algebra, topology and measure theory to explore this idea. Subgroups generated by a set From group theory, arbitrary intersection of subgroups are subgroups. As such for any subset SSS of a group GGG, we can define the subgroup generated by SSS to be ⟨S⟩=⋂H subgroup of G,S⊂HH\braket{S}=\bigcap_{H \text{ subgroup of G}, ...