I have two small questions related to some point-set topology involved in the following. My question is inspired after reading the proof of Proposition 1.26 of Hatcher.
Suppose I have a space $X$ that is built out of a subspace $A$ that is path-connected by attaching some $n$ - cells $e_\alpha^n$ for $n \geq 3$ via attaching maps $\varphi_\alpha : S^{n-1} \to A$. Suppose for each $\alpha$ I choose a point $y_\alpha \in D_\alpha$. Now suppose I set
$U = X - \left\{\bigcup_{\alpha} y_\alpha\right\}.$
I am interested to know why $(1)$ $U$ is open in $X$. It could be that a union of infinitely many point sets is not closed, from which we cannot conclude that $U$ is open. How can I see this fact?
My second question is somewhat related to the proposition as well in that I believe:
$U$ is homotopy equivalent to my subspace $A$ from which $X$ is built out of.
Now I don't know if this is true and I am trying to prove it. I already have an inclusion map $i : A \to U$ to use. However, I don't necessarily have a map going the other way from $U$ to $A$. Is my belief true and if it is, how would I go along proving it? At the end of the day, I would like to be able to fill in the details of this by myself.
Thanks.
Edit: I should say that my $\alpha$'s run over an arbitrary index set, not necessarily countable or anything.