Kleene star and Kleene plus
$\Sigma$ is a finite set of elements called letters. A word is a sequence of 0 or more letters. For every alphabet, there is a unique 0-letter word called the empty word, denoted $\epsilon$. By definition, $\epsilon\notin \Sigma$, for eny $\Sigma$.

Thus, $\Sigma^*$ and $\Sigma^+$ are both well defined, and are used for different purposes.

Mathematically, $\Sigma^*$ with the concatenation operation is a monoid, with $\epsilon$ being the identity element. $\Sigma^+$ is thus a semi-group, with the concatenation operation, as it doesn't have an identity element.