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The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I don't even know where to begin looking for the meaning of a symbol whose latex code is "\Subset". Do you know what this usually denotes?

Edit: some context follows.

All the sets in question are subsets of $\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}$.

Example 1. In a situation where $J$ is closed with empty interior, $U$ and $V$ are closed with $U\subsetneq V$, it is written "Note that $J \Subset U$ and, selecting a neighborhood $W \subset U$ of $J$ which is compactly contained in $V$, ..."

Example 2. In a situation where $R$ is a rational mapping, and where it is assumed that $B\subset \hat{\mathbb{C}}$ is such that $R(B)\Subset B$, it is written "Let $\Omega_0 = \hat{\mathbb{C}}\setminus B$. Define $\Omega_1 = R^{-1}(\Omega_0)$. By the properties of $B$, we have $\Omega_1\Subset\Omega_0$. If we let $U_0$ be any finite union of closed balls such that $\Omega_1 \subset U_0 \subset \Omega_0$, ..."

In both cases I have paraphrased to simplify the notation, so I hope I have not introduced errors into it.

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I have seen $A \Subset B$ mean that the (topological) closure of $A$ is contained in $B$, but I'm sure there are plenty of other uses as well. Could you perhaps provide some context? –  Jesse Madnick Oct 28 '10 at 7:34

1 Answer 1

up vote 7 down vote accepted

In my experience U \Subset V means that the closure of U is a compact subset of V.

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It is much more common, in the sort of papers I read, to see $U\subset\subset V$ for that purpose. However, I find that ugly and have often wanted to use $U\Subset V$ instead. But I have never been sure if that is the recognized use for the symbol. –  Harald Hanche-Olsen Oct 28 '10 at 9:11
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$\subset \subset$ is indeed not very pleasing to the eye, but I always understood it to be a form of $C. \subset$ with $C.$ standing for "compact", which would not be so readily recognized if one used $\Subset$. –  Thierry Zell Oct 28 '10 at 13:23
    
And we read $U \Subset V$ as: "$U$ is compactly contained in $V$. –  Gerald Edgar Oct 28 '10 at 15:58
    
Taking this answer as a starting point and comparing it with the examples I just added, it seems a related possibility for $U\Subset V$ might be "$U$ has a neighborhood which is compactly contained in $V$". Has anyone seen it used this way? –  Linda Brown Westrick Oct 28 '10 at 18:31
    
It sounds like this answer is the consensus. Thanks all! –  Linda Brown Westrick Oct 30 '10 at 19:30

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