-We work in set-theory without the axiom of choice, ZF.
-We answer some questions raised by Johnstone (about almost maximal ideals), see [J84] , and van Douwen (about topology of linear orders), see [D85] .
-We also give some new equivalents of the axiom of choice and of weak forms of the axiom of choice.
-Using existing models of ZF, we obtain some independence results; for example, the following axiom:
The product of non-empty compact Hausdorff spaces is not empty.
does not imply the following axiom of Tychonov:
The product of non-empty compact Hausdorff spaces is compact.
-Our tools are both topological (product of topological spaces), set-theoretic (transfinite recursions) and model-theoretic (reduced-products of first order structures, alias non-standard analysis). See [Mo86], [Mo87], [Mo88a] .