= Addition on the Stone-Čech compactification of the natural numbers
{title2=$+$ on $\beta\mathbb N$}
For ultrafilters $\mathcal U,\mathcal V\in\beta\mathbb N$, define $\mathcal U+\mathcal V$ by
$$
A\in\mathcal U+\mathcal V
\quad\Longleftrightarrow\quad
\{x:\{y:x+y\in A\}\in\mathcal V\}\in\mathcal U.
$$
This operation is <associative operation>[associative], and each right translation $\mathcal U\mapsto\mathcal U+\mathcal V$ is continuous, so $\beta\mathbb N$ is a compact Hausdorff left-topological <semigroup>.
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