# What are these compact sets called?

I'm wondering if a compact set $A\subset\mathbb{C}$ satisfying the properties that

• $A$ and its complement have finitely many connected components

• every connected component of $\partial A$ is the image of a path (say piecewise analytic, or $C^1$)

has a standard name? I'm not sure "set with rectifiable boundary" qualifies, but maybe I'm mistaken. In any case I'm looking for a shorter term if at all possible.

Thank you in advance for any pointer/suggestion.

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How about just a "piecewise smooth set"? –  Lee Mosher May 16 '13 at 11:58
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## 2 Answers

What about "set with piecewise smooth boundary"?

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Alright, but I was looking for something shorter and "more catchy". It is a term I intend to use a lot in an article and therefore I would like to avoid cumbersome expressions as this one. Yet, if nothing better comes up I'll have to face the truth ;) –  Loïc Teyssier May 16 '13 at 10:11
You can define these sets at the beginning of your paper as "good sets", or "allowable sets", or even "regular sets", or something similar. I don't think there is a specific standard name in literature. –  Daniele Zuddas May 16 '13 at 10:14
Yeah, that's what I meant by "the truth" ... Something more poetic would have done, though! –  Loïc Teyssier May 16 '13 at 11:58
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I'll finaly settle for "a cutout compact set" for want of a better term. But I think this word expresses well the "finitely many" (connected components, non-smooth points) side of the object, which is what I wanted to highlight.

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