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We have a uncountable set $S$. Does there exist a surjective map $f$ for any $n \in \mathbb{N}$ and any $x_i \in S$ such that:

$$A:=S-\{x_1,x_2,x_3,\ldots,x_n\}$$

$$f: A \to S$$

The countable case is easy because countability gives you bijections to $\mathbb{N}$. For most examples like $\mathbb{R}^n$ one can use the extra structure on those objects as well. But my problems lies in the generality of the question.

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Let $N$ be any countable subset of $S$ containing $X = \{ x_1, \ldots , x_n \}$. Then define $f$ piecewise: on $S \setminus N$ let it be the identity, and on $N \setminus X$ let it be given by a map $g : N \setminus X \to N$.

This $f$ is surjective if and only if $g$ is surjective, so this reduces the problem to the countable case.

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