Connection number

The cardinality of the family of connectivity components in a topological space. For example, if from the real line one removes $n$ points $a_1<\dots<a_n$, then the components of the remainder are the sets

$$(-\infty,a_1),(a_1,a_2),\ldots,(a_n,\infty),$$

and so the connection number is $n+1$.

The term "connection number" is also used in the following sense. A domain in a Euclidean space is called $n$-connected if its boundary consists of $n$ disjoint connected subsets. For example, the interior of a disc is a $1$-connected domain, the interior of an annulus is $2$-connected.