Exercise 8, § 6

a) Let $\Phi$ be a set of topologies on a set $X$. Show that the neighbourhood filter of a point $x \in X$ with respect to the intersection of the topologies of $\Phi$ is coarser than the intersection of the neighbourhood filters of $x$ with respect to the topologies belonging to $\Phi$.

b) Let $\mathfrak{T}_1$ denote the topology $\mathfrak{T}_0(\mathbf{Q}^2)$ ($\S$ 2, Exercise 7) where the set $\mathbf{Q}^2$ is linearly ordered by the lexicographic ordering (Set Theory,

Chapter III, , no. 6). Let $\mathfrak{T}_2$ denote the topology on $\mathbf{Q}^2$ obtained by transporting $\mathfrak{T}_1$ by means of the symmetry $(\xi, \eta) \to (\eta, \xi)$. Show that if $\mathfrak{T}$ is the intersection of the topologies $\mathfrak{T}_1, \mathfrak{T}_2$, then the neighbourhood filter of a point of $\mathbf{Q}^2$ with respect to $\mathfrak{T}$ is strictly coarser than the intersection of the neighbourhood filters of the point with respect to $\mathfrak{T}_1$ and $\mathfrak{T}_2$