Using the axiom of dependent choice in proofs

[steenis]

I learned something new today: the “Axiom of Dependent Choice”:

The axiom can be stated as follows: For every nonempty set $X$ and every entire binary relation $R$ on $X$, there exists a sequence $(x_n)_{ n \in \mathbb{N} }$ in $X$ such that $x_nRx_{n+1}$ for all $n \in \mathbb{N}$. (Here, an entire binary relation on $X$ is one where for every $a \in X$, there exists a $b \in X$ such that $aRb$.)

See Wikipedia: https://en.wikipedia.org/wiki/Axiom_of_dependent_choice

I want to ask here: what is your experience with this axiom? Did you ever use the “Axiom of Dependent Choice”, how and why?

 

[S.G. Janssens]

Thank you for this.

I am rarely directly confronted with AC and its weaker siblings such as DC, but reading the link made me realize that, yes, I must have been using DC implicitly, because DC is enough to prove Baire's theorem. (In fact, per this reference in the article, DC and Baire are equivalent.) Now, since the metric space version of Baire is used in three foundational theorems in functional analysis (open mapping, closed graph and Banach-Steinhaus), I have indeed implicitly used DC when I was invoking these theorems.

Surely someone closer to the foundations has a deeper insight to offer.