Find the component of $y\in \prod_{\alpha}Y_{\alpha}$

Question

Find the component of $y\in \prod_{\alpha}Y_{\alpha}$

My work

I know the component of $x$ is the union of all subset connected contain $x$

I don't have a clear idea of how attack this exercise. Can someone help me?

Answer 1

The product of connected spaces is connected, so if $y_\alpha$ has connected component $C_\alpha(y_\alpha) \subseteq Y_\alpha$, for all $\alpha$ we have that

$$C(y):= \prod_\alpha C_\alpha(y_\alpha)$$ is connected. And if $C$ is any connected set that contains $y$, $\pi_\alpha[C]$ is connected and contains $y_\alpha$ so by maximality of components :

$$\forall \alpha: \pi_\alpha[C] \subseteq C_\alpha(y_\alpha)$$ which implies

$$C \subseteq C(y)$$ We can conclude that $C(y)$ is the connected component of $y$ in $\prod_\alpha Y_\alpha$.