Glossary of terms

We list a few notations, definitions and statements used in the main text. For what still remains unexplained or unproved we refer to the usual textbooks on elementary analysis, topology and algebra.

Sets

Let Y, Z be sets. If Z ⊂ Y then Y\Z: = {y ∈ Y : y ∉ Z}. The product set Y × Z equals {(y, z) : y ∈ Y, z ∈ Z}. We write Y2 : = Y × Y, Yn+1 : = Yn × Y (n ∈ ℕ, n ≥ 2). Y is countable if there exists a surjection ℕ → Y, otherwise Y is uncountable. The cardinality of Y is strictly less than the cardinality of Z if no map Y → Z is surjective. A partition of Y is a covering of Y by mutually disjoint subsets. The classes of an equivalence relation ∼ on Y form a partition of Y. The set of these classes is denoted Y/∼. The quotient map π : Y → Y/∼ sends each element x ∈ Y into its class π(x). A (full) set of representatives in Y of ∼, or a (full) set of representatives in Y modulo ∼ is a subset R of Y such that π maps R bijectively onto Y/∼.

Let Y = (Y, ≥) be a partially ordered set. A maximal element of Y is an element y ∈ Y such that z ∈ Y, z ≥ y implies z = y.