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I am reading something on Sobolev spaces.

We define $D(\Omega)$ to be the set of function in $C^{\infty}(\Omega)$ that has compact support in $\Omega$.

I know that compact support is completion of the set where a function is non-zero.

but, what does it mean by "have compact support in $\Omega$"? Does it mean it is a function such that its compact support is a subset of $\Omega$?

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Yes, that's exactly what it means. These are functions which are only nonzero on (pre)compact subsets of $\Omega$. –  Neal Oct 16 '12 at 16:09

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I'll turn my comment into an answer. If $\Omega\subset\mathbb{R}^n$ is a domain, a function $f$ on $\mathbb{R}^n$ "has compact support in $\Omega$" provided $supp(f)$ is a compact subset of $\Omega$. Functions with compact support in $\Omega$ are those which are only nonzero on precompact subsets of $\Omega$.

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You can define support of a function by $$supp(f)=\overline{\{x\in\Omega: f(x)\neq 0\}}$$

Now when you say that the function have compact support, you mean that the set $supp(f)$ is compact.

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Maybe I did not make this clear enough, I wanted to know what it meant by have "compact support in $\Omega$". The "in $\Omega$" bit is the key. It has been answered already. Don't worry –  Lost1 Oct 16 '12 at 18:37

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