Exercise 5, § 11

a) Give an example, in the real plane $\mathbf{R}^2$, of a decreasing sequence $(A_n)$ of connected sets whose intersection is not connected (cf. Chapter II, § 4, Exercise 14).

b) In $\mathbf{R}^2$, let $X$ be the set which is the union of the open half-planes $y > 0$ and $y < 0$ and the points $(n, 0)$ where $n$ is an integer $> 0$.

X is connected with respect to the topology $\mathfrak{T}_0$ induced by that of $\mathbf{R}^2$. Define a sequence $(\mathfrak{T}_n)$ of topologies on X such that $\mathfrak{T}_{n+1}$ is finer than $\mathfrak{T}_n$, and such that X is connected with respect to each $\mathfrak{T}_n$ but not connected with respect to the topology which is the least upper bound of the sequence $(\mathfrak{T}_n)$ [pass from $\mathfrak{T}_n$ to $\mathfrak{T}_{n+1}$ by introducing new neighbourhoods of the point $(n + 1, 0)$].*