Behaviour of positive function on compact sets Let $v$ be any strictly positive bounded function on $\Omega \subset \mathbb{R}^n$ and zero on $\partial \Omega.$ 
Can we say that for each compact subsets (w.r.t usual topology defined on $\mathbb{R}^n $) of $\Omega$ there exists a constant $c$ such that $v \geq c >0$?
If not, what can be the example?

Let $n=1$, $\Omega =(-1,1)$, $u(\frac 1 n)=\frac 1 n$ for $n=1,2,...$ and $u(x)=1$ for all other $x \in \Omega$, $u(x)=0$ for $x =\pm 1$. Consider the compact set $[0,\frac  1 2]$. 
