Let $\pi:X \to \Delta$ be a proper, surjective, flat morphism (here $\Delta$ is the unit disc), smooth over $\Delta \backslash \{0\}$ and possibly singular central fiber. There is a fibrewise retraction $r:X \to X_0$, where $X_0$ is the (central) fiber over $0$. Is it true that for a general fiber $X_t$ of $\pi$, the resulting specialization map $$X_t \hookrightarrow X \xrightarrow{r} X_0$$ is surjective? Is it also going to be a homeomorphism?
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1$\begingroup$ If the central fibre is singular it definitely need not be homeomorphism (I assume we are talking about the analytic topoology). For example, a family of smooth conics degenerating to a pair of lines. $\endgroup$– Lazzaro CampeottiCommented Mar 30, 2017 at 10:37
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$\begingroup$ It may not be directly related to your question but it may helpful:Let $X$ be a complex analytic space and let $\pi:X\to \Delta$ be a proper complex analytic function with non-singular generic fibers. By using the blow-analytic methods of Kuo we can construct a retraction of a neighbourhood of the central fibre onto central fiber $\endgroup$– user21574Commented Jul 27, 2017 at 5:15
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$\begingroup$ @potentiallydense surjectivity seems very reasonable. Do you see a proof? $\endgroup$– ArrowCommented Mar 22, 2020 at 15:19
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