Are all closed star-shaped domains in $\mathbb{R}^d$ with d-dimensional interior, regular closed sets? I know all closed star-shaped domains have a star-shaped interior and all d-dimensional open star-shaped domains $\mathbb{R}^d$ are diffeomorphic to $\mathbb{R}^d$ itself. Is that enough to conclude that they are regular closed sets or am I missing something?
2026-03-25 02:57:48.1774407468
Star domain are regular
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Here's a counterexample. Let $d = 2$, and consider the union of the closed disk of radius $1$ with the $x$-axis. Taking the closure of the interior leaves you with only the closed disk.