Are Knots closed?

Question

In every definition I see, a (classical) knot is an embedding of $S^1$ in $S^3$ or $\mathbb{R}^3$. But my lecturer said that the complement of a knot in $S^3$ is open, hence the knot is closed. But I'm not able to prove the fact that a knot should be closed, because embeddings are not required to be closed.

Am I missing something? Is it by definition closed?

Answer 1

The statement

The complement of a knot in $S^3$ is open, hence the knot is closed

means

Given a knot $k$, which is by definition an embedding $k\colon S^1\to S^3$, the complement of the image $S^3\setminus k(S^1)$ is an open subset of $S^3$, hence the image $k(S^1)$ is a closed subset of $S^3$.

Your mistake is thinking that the word "closed" refers to the map $k\colon S^1\to S^3$ being a closed map. Not only should it be clear that a statement of the form

The complement of ____ is open, hence ____ is closed

refers to a subset of a topological space, it doesn't make any sense if you try to fill in the blank with a function.

So, to summarize: it is quite standard for "a knot $k$" to refer either the map $k\colon S^1\to S^3$ itself, or its image $k(S^1)$, interchangeably and without notice. Use context to figure out what is intended.