# Gaps >[!info] Definition > Given $a, b \in 2^\omega$, *$a$ is eventually dominated by $b$*, denoted as $a<* b$, if there exist only finitely many $n$ such that $a(n) \geq b(n)$. > > Let $A=\left\langle a_\alpha: \alpha<\omega_1\right\rangle$ and $B=\left\langle b_\beta: \beta<\omega_1\right\rangle$ be sequences in $2^\omega$. The pair $(A, B)$ is called a *pregap* if for any $\alpha<\alpha^{\prime}<\omega_1$ and $\beta<\beta^{\prime}<\omega_1$ we have $a_\alpha<^* a_{\alpha^{\prime}}<^* b_{\beta^{\prime}}<^* b_\beta{ }$. > If $c \in 2^\omega$ is such that $a_\alpha<^* c<^* b_\alpha$ for all $\alpha<\omega_1$, then we say $c$ *separates* the pregap $(A, B)$. If no such $c$ exists, the pregap ( $A, B$ ) is called a *gap*. > A gap is called *destructible* if there exists an $\omega_1$-preserving forcing that adds a real separating it. > The gap is called *indestructible* if it is not destructible.