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I would like to know how Stone-Čech compactification works with simple examples, like $(0,1)$, $\mathbb{R}$, and $B_r(0)$ (the open ball of $R^2)$. I've studied the one-point compactification and this is way more difficult to understand. All the texts I've found till now start immediately with functions and closure of functions. If somebody could give me some "visible" ideas, I would bear those in mind and understanding the theory would be a little easier.

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There are spaces like $[0,\omega_1)$ with the order topology where the Stone-Cech compactification is the same as the one-point compactification.

On the other hand, the Stone-Cech compactification of the natural numbers $\mathbb N$ has cardinality $2^{\mathfrak c}$. So the answers depends a lot on the space.

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