I was trying to solve the problem: Is $[0,1]$ a countable disjoint union of closed sets? I find a theorem which is very interesting:
Theorem (Sierpiński). If a continuum $X$ has a countable cover $\{X_i\}_{i=1}^{\infty}$ by pairwise disjoint closed subsets, then at most one of the sets $X_i$ is non-empty.
Can anyone please suggest me some books on Real Analysis or Point Set Topology, which contains theorem of this type?
Two books that have this result are Topology. II by Kuratowski (1968) (I don't know the precise location, but the very end of this question says where it's located in the 1950 French edition, which should help you find where it's located in the 1968 English edition) and Set Theory by Hausdorff (English translation of 1935 3rd edition, end of §29.4, pp. 185-186).
Two books that contain many classical point set theory results, but not the Sierpiński result itself (as far as I can tell), are Set Theory. With an Introduction to Real Point Sets by Dasgupta (2014) and Point Set Theory by Morgan (1990).
