Can a subset of $\mathbb{R}^n$ be homotopy equivalent to $S^n$?

Can a subset of $\mathbb{R}^n$ be homotopy equivalent to $S^n$? I am pretty certain the answer is no and I suspect it might be provable using homology groups, but I do not see how. Note that, because $S^n$ is the one-point compactification of $\mathbb{R}^n$, this question is equivalent to asking whether $S^n$ is homotopy equivalent to one of its proper subsets.

Edit: By freakish's comment, the answer is no when the subset is a manifold. However, I am now less certain whether the answer is no for any subset by the example Tyrone showed.

Edit 2: As Maxime Ramzi pointed out, the answer is also no if $X$ is compact by Alexander duality.

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