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{\Large\bf\sffamily Topología}

Segundo Cuatrimestre 2015
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{\sffamily  Ejercicio 3: Conexión y arcoconexión}

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%Sean $\{X_{\alpha}\}_{\alpha\in \Lambda}$ espacios topol\'ogicos.
\begin{enumerate}[a)]
\item Sean $\prod X_{\alpha}$ con la topolog\'ia producto,  $x= (x_{\alpha})\in X$. Pruebe que $C_{x}=\prod C_{x_{\alpha}}$.

\item Sea $\mathbb{R}^{\omega}$ con la topolog\'ia caja. Pruebe que $x\in \mathbb{R}^{\omega}$ está en la misma componente conexa que la sucesión  nula $\mathbf 0$ si y sólo si $x$ es eventualmente cero.\footnote{\textit{Sugerencia}: Construya un homeomorfismo entre $\mathbb{R}^{\omega}$ que mande a $\mathbf 0$ y $x$ a 
una sucesión acotada y una no acotada respectivamente.} Deduzca que  $x,y\in \mathbb{R}^{\omega}$ están en la misma componente conexa si y sólo si $x-y$ es eventualmente cero.

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