Lemma 4.5.8D878F27BAB16

Proof. Compact-support functions separate interior points and recover their original neighborhoods by positive bumps, so the evaluation map is an embedding. Each bump's nonzero set in the closure lies in the image of its compact support; thus the interior image is open and its complement is the boundary. All compact-support coordinates vanish on the boundary, so the remaining coordinates separate its points. For any other such compactification the coordinate extensions define a continuous injection into the same product, with exactly this closure as image by density. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism, proving uniqueness fixing $\Omega $. $\square$