Exercise 8, § 11

In the example of a topology on a product set $X = \prod_{i \in I} X_i$ given in § 4, Exercise 9, suppose that I is infinite and the cardinal t uncountable. Show that if the $X_i$ are connected, regular, and have each at least two distinct points, then the connected component of a point $a = (a_i)$ of X is the set of $x = (x_i)$ such that $x_i = a_i$ except for a finite number of indices $i \in I$. (Reduce to the case $I = \mathbf{N}$; consider two points $b = (b_n), c = (c_n)$ of X such that $b_n \neq c_n$ for all n. For each n let $(V_{nm})_{m \geq 0}$ be a sequence of open neighbourhoods of $b_n$ in $X_n$ which do not contain $c_n$ and are such that $\overline{V}_{n,m+1} \subset V_{nm}$. Let Z be the set of points $(x_n)$ of X with the following property: there exists an integer $k > 0$ such that $x_n \in V_{n,n-k}$ whenever $n \geq k$. Show that Z is both open and closed in X).