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What does the degree symbol in "Let $(K_n)$ be a sequence of compact subsets such that $K_n \subset K_{n+1}^{\circ}$ mean?

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Almost certainly, "interior". – Arturo Magidin May 6 '12 at 23:40
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The interior of the set. – David Mitra May 6 '12 at 23:41
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@ArturoMagidin You or David should post this as an answer so that the asker can accept it. – Alex Becker May 7 '12 at 0:20

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In this context, it almost certainly means the interior of the set.

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