Two problems with set theory

[mmskettios]

Pardon the format--I've never really done this before and I've not ever touched LaTeX.

1)  Let A,B be subsets of a metric space S such that B is open and cl(A) ^ B is nonempty, with ^ being the intersect, and cl(A) is the closure of A.  Show that A ^ B is nonempty.

2)  Let A,B be subsets of a metric space S.  Show that A is open if and only if cl(A

Comments

[phlebas]

1. I'd start from the definition of cl(A).
Once you've solved part 1, does the reasoning from that suggest an approach for part 2?

[mmskettios]

Yes, got it. I can't believe I didn't make that connection between 1 and 2 (probably because I've messed myself up in thinking if a problem appears on entirely unrelated things, they're apparently unrelated). After #1, #2 fell right on through.

[the_s3ntinel]

(Nitpick: This isn't exactly "set theory". Set theory is about infinite ordinals and cardinals, the continuum hypothesis and so forth. This is just basic properties of metric spaces.)

(1) You don't actually need S to be a metric space here - any topological space will do: If B is open and A ∩ B is empty then A ⊆ S - B. Therefore cl(A) ⊆ cl(S - B) = S - B, since S - B is closed. Therefore, cl(A) ∩ B is still empty. But this isn't the proof they want - they want you to use the metric. As the other commenter says, you just need to unpack the definitions. Try to use the fact that x ∈ cl(A) iff ∀ε > 0 ∃y ∈ A such that d(x,y) < ε.

[mmskettios]

Got it. That helped me perfectly, thanks!