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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. – Zhen Lin Dec 15 '11 at 5:57
Do you know how to show that each $X_K$ is connected? – Dylan Moreland Dec 15 '11 at 6:33
This looks like homework. Please read…;. – Nate Eldredge Dec 15 '11 at 7:05

HINT: Let $X$ be a space.

  1. If $\{C_\alpha:\alpha\in J\}$ is a family of connected subsets of $X$ such that $\displaystyle\bigcap_{\alpha\in J}C_\alpha\ne\varnothing$, then $\displaystyle\bigcup_{\alpha\in J}C_\alpha$ is connected.

  2. If $C\subseteq X$ is connected, so is $\operatorname{cl}_XC$.

Depending on what results you already have available, you may need to use (1) more than once.

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