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From a topology standpoint, $\sqrt{\mathbb{R}}$ would be a space $Y$ such that $Y x Y$ is homeomorphic to $\mathbb{R}$. Such a space cannot exist as $Y$ would be connected (as an image of a connected space) hence a singleton or an interval. And an interval times an interval has non-cutpoints (removing such a point leaves the space connected) while $\mathbb{R}$ has none. One can also show that no odd power of $\mathbb{R}$ has such a square root, IIRC.