Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain

Let $n\in$ $\mathbb{N}$. We call a subset $\Omega$ of Euclidean space $\mathbb{R}^n$ an admissible domain if either $\Omega$ is an open subset of $\mathbb{R}^n$, or $\Omega$ is a subset of the closed upper half-space $H^n$ that is open in $H^n$ in the sense of that definition. In the first case set $D=\mathbb{R}^n$, and otherwise set $D=H^n$; we call $D$ the ambient set of $\Omega$.

A differential $n$-form on $\Omega$ is an assignment $\omega$ which to each point $x\in\Omega$ assigns an alternating $n$-linear form $\omega_x$ on $\mathbb{R}^n$. For each $i\in\{1,\dots,n\}$ let $e_i\in\mathbb{R}^n$ denote the $i$th standard basis vector, whose $i$th coordinate is $1$ and whose other coordinates are $0$. The coefficient function of $\omega$ is the function

$$f:\Omega\to\mathbb{R},\qquad f(x)=\omega_x(e_1,\dots,e_n).$$

We say that $\omega$ is continuous if $f$ is continuous at every point of $\Omega$.

The support of $\omega$, denoted $\operatorname{supp}\omega$, is the set of all $x\in D$ such that every open subset $W\subseteq\mathbb{R}^n$ with $x\in W$ contains a point $y\in W\cap\Omega$ with $\omega_y\ne 0$. Let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are open in the metric space $(\mathbb{R}^n,d_E)$ with the Euclidean distance $d_E$, which is a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence; $\mathcal{T}_{d_E}$ is a topology on $\mathbb{R}^n$ by The Open Subsets of a Metric Space Form a Topology. We say that $\omega$ is compactly supported in $\Omega$ if $\operatorname{supp}\omega$ is compact in $(\mathbb{R}^n,\mathcal{T}_{d_E})$ and $\operatorname{supp}\omega\subseteq\Omega$.

The zero extension of the coefficient function of $\omega$ is the function $\tilde f:D\to\mathbb{R}$ defined by $\tilde f(x)=f(x)$ for $x\in\Omega$ and $\tilde f(x)=0$ for $x\in D\setminus\Omega$.