Well-ordering meaning

A total ordering is called a well-ordering if, in addition, every non-empty subset of X has a minimal element.

The council are empowered to make by-laws for fixing the number of lodgers and separating the sexes therein, promoting cleanliness and ventilation, giving of notices and taking precautions in case of any infectious disease, and generally for the well-ordering of such houses.

His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction.