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The Stone–Čech compactification $\beta((0, 1))$ of $(0, 1)$ can be described as the closure of $(0,1)$ in $\prod_{j\in J} \mathbb{R}$ where $J$ is the set of bounded continuous functions $f : (0,1) \rightarrow \mathbb{R}$. Also recall that for any compactification $Y$ of $(0,1)$ there is a unique map $\beta((0,1)) \rightarrow Y$ which is the identity on $(0,1)$. Let $Y$ be the topologist's sine curve and give a formula for the image in $Y$ of a point $(x_f)_{f \in J}$ in $\beta((0,1))$.

The compactification takes points, say $0.5$, to $(f(0.5),\dots, f_j(0.5))$, but I'm not really sure how to proceed from there.

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