I have a solution for problem one. However, it can be regarded as an ugly proof because it is set theory and nothing about metric spaces plus I think it uses the axiom of choice.
First note that any singelton element can be expressed as an intersection of infinitely disks skrinking to that point.
Next (key step) if we have a union of an intersection of sets we can write it as an intersection of union of sets. So for example, (this is verified by the distribution law of unions and intersections). However, the problem is what happens if we have infinitely sets, possibly uncountably many, does this still work. I think it still works but the problem is that it is not longer a topology question now it is a set theory question, that is the ugliness of my solution because I change one area in math into another area in math.
But now we can prove it. Say is an non-empty set. Pick any then we can write (a singleton) as where are open disks skrinking to . Pick another point and do the same idea. Now define but here it can be an uncountable intersection , but using the thing that I said above we can write this union of intersections as intersections of union of sets. BUT, the union of open sets is open so we are taking an intersection of open sets. Q.E.D.
(What an ugly construction. )