$(0^n 1)^* \ \ , n\geq 0 $
According to wiki
If V is a set of strings, then V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation
If V is a set of symbols or characters, then V* is the set of all strings over symbols in V, including the empty string ε.
So this language accepts all strings over $\Sigma^*$ which must be regular. Also regular languages are closed under kleene star.
But again on wiki
$V^* = \bigcup\limits_{i\geq 0}^{} V_i = {\epsilon} \ \cup \ V_1 \ \cup V_2 \ \cup \ V_3 \ \cup .....$
Now according to this definition strings such as $01001$ cannot be a part of given language so $0$'s prior of every 1 are compared within a string, so this can't be regular.
But according to the former definition $01001$ is a part of language because it can be formed with symbols $01$ and $001$ both are part of $0^n 1$.
Can someone help me in identifying the class of these types of languages