TITLE: If the empty set is a subset of every set, why write ... $\cup \{∅\}$?
QUESTION [6 upvotes]: I met the notation $ S=\{(a,b] ; a,b\in \mathbb R,a<b\}\cup\{\emptyset\} $
I know $S$ is a family of subsets ,a set of intervals, and from set theory $\emptyset$ is a subsets of every set then why in the notation :$ S=\{(a,b] ; a,b\in \mathbb R,a<b\}\cup\{\emptyset\} $ appear $\color{red}{\cup\{\emptyset\}}$?

REPLY [6 votes]: The answer is: the given definition uses $\cup\,\{\emptyset\} $, not $\cup\,\emptyset $, so it adds the empty set as an element, not a subset of $S $.