Closed regular neighborhood

Question

I would like to understand the following sentences.

Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$.

Here what I do not understand is the definition of closed regular neighborhood. It seems that $U$ is solid tori whose cores are $L_i$'s.

I know such solid tori are some kind of neighborhoods, but why is it closed? In my understanding, "closed" means "compact and no boundary". But a solid torus has a boundary.

Also what is "regular" here?

I appreciate any help.

Thank you.

Answer 1

Closed here means "closed in the sense of general topology". (You can also say "compact" if you prefer.)

Unfortunately, the words "closed", "interior" and "boundary" have different meaning in geometric topology and general topology and, moreover, this meaning is frequently mixed (e.g. the general topology meaning is used in geometric topology papers). Fortunately, it is (mostly) clear from the context which meaning is being used. (If something cannot be closed as a manifold then it is closed as a subset.)