Problem on cardinality of path-connected components . 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.