If one has a map $f$ from a topological space $X$ to another space $Y$ and then one takes the compactification of $Y$ (for example, if $Y = \mathbb{R}^n$ the compactification is constructed by taking the point at infinity), does $f^{-1} $ take a point in the compactification of $Y$ back to a point in the compactification of $X$?
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4$\begingroup$ Most maps don't have inverses, and even if $f^{-1}: Y \to X$ exists, there's no guarantee that we can extend it to $Y^*$ and maintain continuity. $\endgroup$– Sammy BlackCommented Dec 1, 2023 at 21:34
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1$\begingroup$ Which compactification? There are infinitely many possible compactifications. What you describe for $\Bbb R^n$ is not called "the compactification", but the one-point compactification, because it is accomplished by adding a single point to $\Bbb R^n$. $\endgroup$– Paul SinclairCommented Dec 2, 2023 at 23:37
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