Munkres provides us with this lemma before performing the proof.
This proof is very clear; then, we are presented with the following theorem, with which I am having some difficulties.
My first question is, what topology do the sets $C$ and $D$ belong to? We have not assumed that $Y$ is a subspace, only that the sets in the union are connected subspaces; is it just assumed that we are taking $Y$ as a subspace of $X$?
Similarly, my second question is with respect to the highlighted area. I don't see how this follows from the lemma; $A_\alpha$ has not been specified as being taken in the subspace topology of $Y$, yet the use of the lemma would require $A_\alpha$ be a subspace of $Y$.
I think, perhaps, I have a fundamental misunderstanding of when we are permitted to take subsets as being in the subspace topology. Can we just do it arbitrarily, and have it be implied we are considering certain subsets as subspaces?