Embedding topological manifolds into Lp spaces

With a simple argument, we show as a main note that, for every given 1 ≤ p ≤ +∞, every locally compact second-countable Hausdorff space is topologically embeddable into some L p space with respect to some finite nonzero Borel measure, where the embedding may be chosen so that its range is included in some open proper subset of the Lp space.