Weak conditions on boundary for the existence of smooth cut-functions

Question

Let $\Omega\subset\mathbb R^d$ be bounded and open. What are weak conditions on $\Omega$ that guarantee the existence of functions $\mathbf 1_{K(n)}\in C^\infty(\Omega)$ and compact sets $K(n)\subset \Omega$, $n\in\mathbb N$, such that $\cup_n K_n=\Omega$, and for each $n\in\mathbb N$ it holds that $K(n)\subset K(n+1)$ and $0\le \mathbf 1_{K(n)}\le 1$ with \begin{equation} \mathbf 1_{K(n)}=\left\{\begin{split} &1,\ \ \text{on }K(n)\\ &0,\ \ \text{on }\Omega\backslash K(n+1). \end{split} \right. \end{equation}

Answer 1

$\Omega$ being open is enough. You can construct a sequence of exhausting compact sets as $$ K(n)=\{x\in\Omega:\mathrm{dist}(x,\partial\Omega)\le\frac1n\}. $$ Given $n$, let $d_n=\mathrm{dist}(K(n),\partial K(n+1))>0$, take $0<\epsilon<d_n/2$ and let $$ w(n)=\bigcup_{x\in K(n)}B(x,\epsilon), $$ where $B(x,\epsilon)$ is the ball of center $x$ and radius $\epsilon$. $w(n)$ is an open set such that $$ K(n)\subset w(n)\subset K(n+1). $$ Finally, take your favorite bump function $\phi$ and let $$ \mathbf 1_{K(n)}(x)=\phi_\delta\ast\chi_{K(n)}(x), $$ where $0<\delta<\epsilon$, $\phi_\delta(x)=\delta^{-n}\phi(x(\delta)$ and $\chi_A$ is the characteristic function of $A$.