How could I call the "pieces" of a non simply connected domain?

Question

I am writing an article and I would like to use the formal terminology but I am not aware if there is some. For instance assume that $\Omega$ is a non-simply-connected domain, for instance let $\Omega:=[0,10]\bigcup[20,30]$ . Is there any name I can give the simply connected part of this non-simply connected domain? I could simply rename them as $\Omega_1$ and $\Omega_2$ but in my particular case it would render things complicated because I would like to work in general terms. Thanks in advance.

Answer 1

If you only care about subsets of $\mathbb R$, and if $[0,1)\cup\{2\}\cup[5,\infty)$ has three "pieces", then I would just call the pieces "components". If you wouldn't want to count $\{2\}$ because it has length $0$, then you could call the other two "'proper' intervals" (perhaps alternately "'genuine' intervals"), but some reserve that for bounded intervals only.

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In topology, there are many related concepts: A topological (sub)space could be connected, path-connected, arc-connected, simply-connected. It turns out that for subspaces of $\mathbb R$, all of those concepts coincide. For most of these, we can talk about components that are connected in that way. But we would not speak of "simply-connected components" since simply-connected means something like "has no holes", and which are the no-holes pieces of an annulus?