Remarks on the interaction between pointfree topology and certain choice principles, specifically the Axiom of Choice (AC), the Axiom of Countable Choice (ACC), and the Boolean Prime Ideal Theorem (PIT), in connection with the following issues: (1) The equivalence of PIT with certain spatiality conditions, (2) the relation between the classical StoneCech compactification and its pointfree counterpart (which does not require any choice principle), (3) the same regarding the classical Hausdorff Tychonoff Theorem and the pointfree Tychonoff Theorem (which also does not involve any choice principle), (4) the equivalence of ACC and the Lindelofness of certain 0dimensional frames, and (5) the equivalence of AC and PIT with certain conditions concerning, respectively, the maximal and the prime elements of coherent frames.
