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One point compactification of $(\mathbb{C}^{\ast})^n$

If $\widehat X$ is the 1-point compactification of $X$, then there is a homeomorphism (for, say, locally compact Hausdorff spaces) $$ \widehat{X \times Y} \cong \widehat X \wedge \widehat Y $$ with th …
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