A $3$-manifold $M$ (without boundary) is irreducible if every smoothly embedded $2$-sphere in $M$ bounds a $3$-ball.
What would be a “good” definition of irreducibility for $3$-manifolds with boundary? Is it necessary to add extra requirements?
For example, let $M = \mathbb{R}^3 \setminus B$, where $B$ is an open ball. I would like $M$ to be irreducible. However, if $B’$ is a ball in $\mathbb{R}^3$ that contains $B$, then its boundary does not bound a ball in $M$, so $M$ is not irreducible according to the original definition.