This is inspired by this discussion. I see that the debates about the necessity of the axiom of choice in this or that statement are still ongoing. In this regard, I became interested in whether there are textbooks that describe what mathematical disciplines turn into when the axiom of choice is consistently excluded from them.

It seems to me that, theoretically, nothing prevents a logician from writing a book that describes algebra or topology or analysis in some axiomatic set theory without the axiom of choice, for example, in ZF.

So my question is

are there texts describing what remains of analysis (or algebra, or topology) if it is built consistently without the axoim of choice, for example, in ZF (not in ZFC)?

I think such a text would be very helpful, because in my observation, people who argue about the application of the axiom of choice tend to focus on particulars without seeing the big picture, and moreover, having a rather vague idea of the subject.

For example, I have met analysts who believe that in analysis the axiom of choice appears only in a few statements, such as the Hahn-Banach theorem, and if you do not use them, you can consider your conscience "unstained by its application".

People on this thread have already explained to me that some things in analysis can be preserved, in particular (forgive my ignorance), it was a surprise to me that the theory of real numbers is preserved in some form (although, as far as I understand, the classical theorems of mathematical analysis mostly disappear). And apparently, something can be preserved in algebra and topology. I would be terribly interested to look at this picture.

I remember Boris Kushner's book on "Constructive mathematical analysis", but it's about something else, about a variant of intuitionistic mathematics, where "constructiveness", as the adepts understand it, is woven into the system of axioms of logic, not set theory. Is there any overview or a book explaining what is preserved in mathematics when it is built in ZF?

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