Hjdcti

I met the notation $ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $

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,bin mathbb R,a<bcupemptyset $ appear $colorredcupemptyset$?

It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.

Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$

Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).

The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.

It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.

As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$