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Let $\{X_\beta\}_{\beta \in J}$ be an indexed family of connected spaces, and let $X := \prod \limits_{\beta \in J} X_\beta$ with product topology. Fix $(\alpha_\beta)$ in $X$. Fix a finite subset $K$ of $J$ and let $X_K$ be the subspace of $X$ containing all $(x_\beta)$ in $X$ for which $x_\beta = \alpha_\beta$ if $\beta$ is not in $K$. Are the union $Y$ of all such spaces for all possible finite sets $K$ connected, and does this imply that $X$ is connected?

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    You may insert mathematical formulae on this site by enclosing LaTeX code in `$...$`. Also, please consider phrasing your posts as _questions_ rather than _orders_.2011-12-15
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    Do you know how to show that each $X_K$ is connected?2011-12-15
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    This looks like homework. Please read http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question.2011-12-15

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