# 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
@LoïcTeyssier Very good type of subsets of $\mathbb{C}$. An operator theoretical version of your question:"To what extent all $C^{*}$ algebras with the property that the spectrum of all its element,as a subset of $\mathbb{C}$, satisfies 2 properties you mentioned, are classified? What are examples of infinite dimensional $C^{*}$ algebra with this property? what would be an algebraic (and spectral) interpretation for this topological property which you mentioned? On the other hand every compact set of $\mathbb{C}$ determines an ideal $J$ in $C_{0}(\mathbb{C}$.so the next question could be : – Ali Taghavi May 14 '14 at 8:05
What can be said about the extension $0\to J\to C_{0}(\mathbb{C})\to C_{0}(\mathbb{C})/J\to 0$, provided your topological property is hold? Does this introduce a new $C^{*}$ algebraic invariant? – Ali Taghavi May 14 '14 at 8:08
What can be said about the Busby invariant of such type of Extension? Morover, The Brouwn Douglas Filmore theory is about the classification of essential normal extension of compact operators in $B(H)$. If I am not mistaken it is closely related to the relative position of compact subsets of the plane. So it would be interesting to obtain some interpretation for your property, in term of BDF theory. – Ali Taghavi May 14 '14 at 8:26
@AliTaghavi : I don't know anaything about $C^*$-algebras, sorry... – Loïc Teyssier May 14 '14 at 10:51