Rectifiable spaces
The author studies those spaces \(X\) which are rectifiable, i.e., for which there are a surjective homeomorphism \(\psi : X^2 \twoheadrightarrow X^2\) and a distinguished element \(e \in X\) such that every \(x \in X\) satisfies (a) \(\Psi (x,x) = (x,e)\) and (b) \(\Psi [\{x\} \times X] = \{x\} \times X\). Every topological group is rectifiable and every rectifiable space is homogeneous; both converse statements fail. The thrust of the paper is that several familiar assertions about topological groups remain true for rectifiable spaces. For example, first countability yields metrizability. The author shows further, with \(w, \chi, \pi w, \pi \chi\), \(d\) and \(k\) denoting respectively weight, character, \(\pi\)-weight, \(\pi\)-character, density character and compact-covering number, that the following relations are valid for arbitrary rectifiable spaces \(X : \pi \chi (X) = \chi (X)\); \(w(X) = d(X) \cdot \chi (X)\); \(w(X) \leq k(X) \cdot \chi (X)\), with equality if \(X\) is locally compact; and \(\pi w(X) = w(X)\).
scientific article; zbMATH DE number 858602
scientific article