The $\delta$-Interior of an Open Subset of $\mathbb{R}^n$ is Open

Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$ and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^n$; write $\bar B(x,r)$ for the closed ball of centre $x$ and radius $r$ in $(\mathbb{R}^n,d)$.

Let $\Omega\subseteq\mathbb{R}^n$ be open in $(\mathbb{R}^n,d)$, let $\delta>0$ be a real number, and put

$$\Omega^{\delta}=\{x\in\mathbb{R}^n:\bar B(x,\delta)\subseteq\Omega\}.$$

1. (Room to spare) For every $x\in\Omega^{\delta}$ there is a real number $\eta>0$ with $\bar B(x,\delta+\eta)\subseteq\Omega$.

2. (Openness) $\Omega^{\delta}$ is open in $(\mathbb{R}^n,d)$.