On a connected domain, the set of antiderivatives of a continuous function $f$ is $1$-dimensional. However, for a punctured domain like $\mathbb{R} - \{0\}$, the set of antiderivatives becomes $2$-dimensional. I wonder, what about a domain with countably many holes, like the domain of $tan(x)$? Is it countably-infinite dimensional, or uncountable dimensional?
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