# Tough Connected Problem

• November 16th 2013, 01:04 PM
Aryth
Tough Connected Problem
I just need help understanding this problem:

Suppose that $\{X_i,\Omega_i) : i\in I\}$ is a family of topological spaces and suppose that $\bold{x}$ and $\bold{y}$ are members of $\prod \{X_i : i\in I\}$. If $\bold{x}$ and $\bold{y}$ differ in only one coordinate, prove that they must lie in the same connected component.

What we have to work with:

Let $(X,\Omega)$ and $(Y,\Theta)$ be topological spaces. If $(X,\Omega)$ is connected and the function $f: X \to Y$ is continuous relative to $\Omega$ and $\Theta$, then $f(X)$ is connected as a subspace of $(Y,\Theta)$.

Suppose that $\{X_i,\Omega_i) : i\in I\}$ is a family of topological spaces. Fix $i\in I$; and for all $j\in I - \{i\}$, fix $a_j \in X_j$. Define the function $f_i: X_i \to \prod\{X_i : i\in I\}$ by $f_i(x) = \bold{x}$, where $\pi_i(\bold{x}) = x$ and $\pi_j(\bold{x}) = a_j$ when $j\neq i$. Using this map, it can be shown that $(X_i,\Omega_i)$ is homeomorphic to the subspace $f_i(X_i)$.

Both of these have been proven in class. I'm just failing to see how to use these when one coordinate differs. Any help is greatly appreciated!