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Suppose we have a family of compact, first-countable spaces $K_i$. Let $L$ be the one-point compactification of their disjoint sum. Must $L$ be first-countable?

Note that if we drop the compactness assumption then the answer is no: simply take $K=\omega_1$.

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No. For example, take the $K_i$ to be one-point spaces, and take uncountably many of them.