What is the complement of a loop?

Question

My Algebraic Topology book says

$A$ is a loop in the complement of another loop $B$

What does "in the complement of" mean here?

Answer 1

In topology, the complement of a loop (or a knot, or a link...) is just the set-theoretic complement: $X \setminus A$ where $A$ is the loop and $X$ is the ambient space. The ambient space is meant to be inferred from the context; usually it's either $S^3$ or $\mathbb R^3$.