Why does Garling's theorem 1.2.2 require the separation axiom?

[Math Amateur]

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...

The Separation Axiom and Theorem 1.2.2 read as follows:
View attachment 6137

Garling argues that the Separation Axiom needs to be in place before we can prove Theorem 1.2.2 ... ... but I cannot see where the Separation Axiom is needed in the proof of Theorem 1.2.2 ...

Can someone give a clear explanation of exactly why we need the Separation Axiom in order to prove Theorem 1.2.2.

Help will be much appreciated ... ...

Peter

 

[GJA]

Hi Peter,

Garling argues that the Separation Axiom needs to be in place before we can prove Theorem 1.2.2 ... ... but I cannot see where the Separation Axiom is needed in the proof of Theorem 1.2.2 ...

The application of the Separation Axiom is what justifies the statement "and so there exists a set $b=\ldots$" The set $A$ in the axiom statement is $\Omega$ in the theorem. Does this help?