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It always possible to shrink a domain $U_\alpha$ to $V_\alpha$ in the sense that $$V_\alpha\Subset U_\alpha\quad\text{and} \quad\bigcup_{\alpha}U_\alpha=\bigcup_{\alpha}V_\alpha\,?$$

What does $\Subset$ mean ? Is it an error ?

Alex Shpilkin
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Rick
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  • @NormalHuman: My question is only on the sense of $\Subset$. What other tag can be used excepted "Notation" ? – Rick Nov 28 '15 at 18:42
  • Notation is context-dependent. Presumably this has to do with topology or analysis? –  Nov 28 '15 at 18:43
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    These seem to be relevant: http://mathoverflow.net/questions/43950/meaning-of-subset-notation http://math.stackexchange.com/questions/56872/the-meaning-of-a-notation-from-complex-analysis – Tomas Nov 28 '15 at 18:43
  • You could tell us in what source you found the notation. Have you checked that the author did not define the notation? – mickep Nov 28 '15 at 18:44
  • ^ same thing with less fancy typography. –  Nov 28 '15 at 18:46

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According to my Differential Geometry professor, it means that the closure of $V_{\alpha}$ is contained in $U_{\alpha}$.

According to Silvia Ghinassi and other sources, it generally means that the closure of $V_{\alpha}$ is a compact subset of $U_{\alpha}$, in which case the notation $V_{\alpha}\Subset U_{\alpha}$ is read "$V_{\alpha}$ is compactly contained in $U_{\alpha}$".

MoebiusCorzer
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