What does the notation $\subset \subset$ mean?

In my class notes, our prof writes $\Omega \subset \subset \mathbb{R}^{n}$ to mean that "$\Omega$ is a convex subset of $\mathbb{R}^{n}$". Is that all that this means, or is there any more to it than that? I'm just curious because I've never seen this notation before, and don't want any of it's subtleties or nuances lost on me.

  • $\begingroup$ math.stackexchange.com/questions/56872/… $\endgroup$ – Poppy Jan 26 '14 at 19:00
  • $\begingroup$ That question seems to not come to a consensus as to whether it means "relatively compact" or "bounded " :( $\endgroup$ – ALannister Jan 26 '14 at 19:02
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    $\begingroup$ Relatively compact means "whose closure is compact". Closure is of course always closed :) and if we have boundedness, then we also have compactness (in $\mathbb{R}^n$). $\endgroup$ – Poppy Jan 26 '14 at 19:04
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    $\begingroup$ What I'm trying to say- if we are in $\mathbb{R}^n$, then relatively compact is the same as bounded. $\endgroup$ – Poppy Jan 26 '14 at 19:05
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    $\begingroup$ True. Not something we get in infinite dimensional spaces, unfortunately, but we're not dealing with those here . Thanks for the clarification, Poppy! $\endgroup$ – ALannister Jan 26 '14 at 19:06

The usual meaning is "strongly included": $A\subset\subset B$ when $\overline A$ compact and $\overline A\subset B$.



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