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In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. I never figured out what it means exactly; I always interpreted it as the closure $\overline{\Omega}$ is contained in $\Omega'$ (so that some nasty boundary effects can be avoided). Is that right? Or does $\subset\subset$ mean some other kind of inclusion? Thanks.

Martin R
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  • This is the first time I see this. For me it is a missprint. Let's wait for the specialists. – Sigur Nov 13 '13 at 00:18

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It usually (the only meaning I've come across yet) means that $\Omega$ is relatively compact in $\Omega'$, so the closure of $\Omega$ is compact and contained in $\Omega'$.

Daniel Fischer
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  • $A\subset X$ is said to be relatively compact if the closure of $A$ is compact. Now $\Omega\subset\Omega'$ is relatively compact would mean the closure of $\Omega$ in $\Omega'$ is compact. That is $\overline\Omega\cap\Omega'$ is compact. Why does it follow that $\overline\Omega\subset\Omega'$? – Not Euler Nov 01 '19 at 10:24
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    @HritRoy Since $\mathbb{C}$ is Hausdorff, $\overline{\Omega} \cap \Omega'$ being compact implies it is closed in $\mathbb{C}$, whence $\overline{\Omega} \subset \overline{\Omega} \cap \Omega'$. – Daniel Fischer Nov 01 '19 at 10:29