Exercise 2, § 11

a) Let $X$ be a topological space and $\mathfrak{T}$ the topology of $X$. Show that if $X$ is connected in the semi-regular topology $\mathfrak{T}^*$ associated with $\mathfrak{T}$ (§ 8, Exercise 20), then $X$ is connected in the original topology $\mathfrak{T}$. Deduce that there exist connected submaximal spaces (§ 8, Exercise 22).

*b) In particular construct a topology on the real line $\mathbf{R}$, with respect to which $\mathbf{R}$ is connected but no point of $\mathbf{R}$ has a fundamental system of connected neighbourhoods [cf. Chapter IV, § 2, Exercise 14 c)].