Prove that a retraction is a quotient map

[Jamin2112]

Homework Statement

As in title.

Homework Equations

Described in my attempt.

The Attempt at a Solution

Where do I go from here? I need to show that those 2 unioned sets are open in A. I'm not seeing it

 

[Jamin2112]

Wait ... hang on ... I think I might have it.

I know that (r^(-1) (U) ⋂ A) is open in A if we mean with respect to the subspace topology, since it is the intersection of an open set r^(-1) (U) of X with A. Not sure about the other unioned set though. Thoughts?