The problem is : Does there exist any compact topological space X such that the set of all path-connected components of the space X is exactly aleph-naught ???
Actually, I was thinking about the problem when for a given cardinal number § can we found a space X, satisfying some separation or countable axioms such that the number of (connected/path/quasi) components of X are precisely § ??
I am completely unable to think of any non-trivial topological spaces except discrete topology , and I have tried some known spaces but couldn't find it out . Also, please provide me some reference book(s) on these topic(s) .
For a compact example with countably many path components, take $X = \{\frac{1}{n} \mid n \in \mathbb N\} \cup \{0\}$ is such a space.
For a compact example with any given cardinal number, represent that cardinal number by a well ordered set $X$ that possess a maximal element, and use the order topology on $X$.
If you don't like examples in which the path components are all points, take the cartesian product of any of my examples $X$ with your favorite compact path connected set.