Exercise 4, § 7

a) Let $X, Y$ be two topological spaces, $\mathfrak{F}$ a filter on $X$ having a cluster point $a$, and $\mathfrak{G}$ a filter on $Y$ having a cluster point $b$. Show that $(a, b)$ is a cluster point of the product filter $\mathfrak{F} \times \mathfrak{G}$.

b) In the product space $\mathbf{Q}^2$, give an example of a sequence $(x_n, y_n)$ which has no cluster points although each of the sequences $(x_n), (y_n)$ has a cluster point in $\mathbf{Q}$.