Non-Archimedean f-rings need not be p-distributive. Moreover, if {di|i} is a subset of a non-Archimedean f-ring and a ≥ 0, the elements a vi di and vi adi need not be equal. We prove, however, that the difference is an infinitely small element when the ring has a strong unity.
In this expository article we use topological ideas, notably compactness, to establish certain basic properties of orderable groups. Many of the properties we shall discuss are well-known, but I believe some of the proofs are new. These will be used, in turn, to prove some orderability results, including the left-orderability of the group of PL homeomorphisms of a surface with boundary, which are fixed on at least one boundary component.