How Regularity of $X$ implies $\mathcal {A}$ covers $X$?

Question

[Reference:Munkres Topology $2nd$ edition]

$X$ is PARACOMPACT+ $X$ is HAUSDORFF$\implies $ $X$ is NORMAL$\implies $ $X$ is Regular.

I'm not getting

(1)How does "Regularity of X implies that $\mathcal {A}$ covers X"?

(2) What is the meaning of the phrase "indexing a set bijectively"?

Please help me answering the above questions,any kind of suggestion/hint will be greatly appreciated

thank you!

Answer 1

$\mathscr A$ covers $X$:

Let $x \in X$. There exist $\alpha \in J$ such that $x \in U_\alpha$. By regularity there exists an open $A \subset X$ such that $x \in A$ and $\overline A \subset U_\alpha$. Clearly $A \in \mathscr A$.

Indexing $\mathscr B$ bijectively:

This means that we take any bijection $b : K \to \mathscr B$ from a set $K$ to $\mathscr B$. This "choice step" is not really needed. You can take $K = \mathscr B$ and $b = id$.