Why is $M$ closed in the weak topology of $V = H^{1,2}(\varOmega;\mathbb{R}^N)$?

Question

Let $\varOmega$ be a bounded domain in $\mathbb{R}^n$, and let $S$ be a compact subset in $\mathbb{R}^N$. And also let $u_0\in H^{1,2}(\varOmega;\mathbb{R}^N)$ with $u_0(\varOmega)\subset S$ be given. Define $$H^{1,2}(\varOmega;S) = \{ u\in H^{1,2}(\varOmega;\mathbb{R}^N); u(\varOmega)\subset S \text{ almost everywhere}\}$$ and let $$ M = \{u\in H^{1,2}(\varOmega;S); u-u_0\in H_0^{1,2}(\varOmega;\mathbb{R}^N)\}.$$ Why is $M$ closed in the weak topology of $V = H^{1,2}(\varOmega;\mathbb{R}^N)$?