Show that the regular sets arc not closed under infinite uni

Show that the regular sets arc not closed under infinite union by producing an infinite family of regular sets whose union is not regular.

Solution

Consider the sets {0}, {01}, {0011}, etc.

Each one is regular because it only contains one string. But the infinite union is the set {0i1i | i>=0} which we know is

not regular. So the infinite union cannot be closed for regular languages.