Meaning of $\Subset$ notation

Question

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.

Answer 1

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

Answer 2

A. I think $U \Subset V$ and $U\subset\subset V$ mean that $U\subset K\subset V$ for some compact $K$, i.e., "$U$ is compactly contained in $V$".

B. In a Hausdorff space, this is equivalent to "$\overline U\subset V$ and $\overline U$ is compact".

C. If $V$ is locally compact Hausdorff, this is equivalent to "$U$ has a neighborhood which is compactly contained in $V$".

Note: open and closed subsets of $\mathbb R^n$ are locally compact. https://en.wikipedia.org/wiki/Locally_compact_space

Proofs: B.: If "$\overline U\subset V$ and $\overline U$ is compact", we can take $K:=\overline U$, so assume $U\Subset V$ in a Hausdorff space. Then $K$ is necessarily closed, by http://mathonline.wikidot.com/compact-sets-in-hausdorff-topological-spaces Always a closed subset of a compact set is compact (proof: add its complement to get an open cover of the latter set); hence so is $\overline U$. As $K$ is closed, $\overline U\subset K\subset V$.

C. $V$ being locally compact means that every point has a compact neighborhood, i.e., $x\in W_x\subset K_x$, where $W_x$ is open and $K_x$ compact.

As $K$ is compact, $K\subset W_{x_1}\cup\cdots\cup W_{x_n}=:W$ for some $x_1,\ldots,x_n\in K$. But $W\subset K_{x_1}\cup\cdots\cup K_{x_n}=:K'$, which is compact in $V$ (hence closed); hence so is $\overline W\subset K'$. Thus, $U\subset W\subset \overline W\subset V$, QED.

Note: B. is the definition used in the other answer and C. answers Linda's comment to it.