In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF or at most ZF+ADC.

PS: ZF<ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach