1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X? 2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set) 3. If 2 is true, can we use De Morgan's law to show than 1 is true? 4. If 1 is true, why do we require "countable unions" is the definition of sigma algebra?