Looking for some guidance on two topology questions:
(a) Show that a locally connected space with a countable basis, has at most
countably many connected components.
(b) Give an example when X has countable basis but it has uncountable many
connected components.
Mainly stuck on (b). I think I understand why this is the case for (a) along the lines that for $d$ in a set $D$ (with a countable basis), given $C_{d}$ a connected component for $d$, the intersection between the set of such countable components, $C$, and $D$ is non-empty i.e. $C \cap D \neq \emptyset$, but this means that, by the countability of $D$'s basis, we cannot have uncountably many connected components (elements in the intersection). For (b) I've been casting about and have seen some references in texts to sets $B_{\rho,\theta}$ which have uncountably many connected components but countably many components that are copies of the Mandelbrot set…but suspect this may be over-complicating.
Best Answer
As to (1), let $D$ be a countable dense subset of $X$. In a locally connected space $X$, all components $C_x$ of a point $x \in X$ are open sets, and different components are disjoint.
Every distinct component contains a point of $D$ (as components are open and $D$ is dense), and this defines an injective function from the set of all different components into $D$, so the set of components is at most countable.
The irrationals or the Cantor set show that there are second countable, uncountable spaces, where the set of components is all singletons.