Some addition theorems for rectifiable spaces (Q2883529)

A space \(X\) is said to be \textsl{rectifiable} if there is \(e\in X\) and a homeomorphism \(g:X\times X\to X\times X\) such that for each \(x\in X\), \(g((x,e))=(x,x)\) and \(g\) restricted to the subspace \(X_x=\{(x,y):y\in X\}\) is a (surjective) autohomeomorphism. Every topological group is rectifiable and all rectifiable spaces are homogeneous. The purpose of this paper is to establish addition theorems for rectifiable spaces. The main theorem of Section 2 states that if \(B\) is a compact Hausdorff space of cardinality less than \(2^{\omega_1}\), and \(B=X\cup Y\), where \(X\) and \(Y\) are non-locally compact rectifiable spaces, then \(B\), \(X\) and \(Y\) are separable and metrizable. Another result of this section states that if (compact Hausdorff) \(B\) has countable tightness and can be expressed as a countable union of dense rectifiable subspaces, then each of these (and \(B\)) is separable and metrizable. Section 3 deals with \(k\)-gentle paratopological groups, that is to say, paratopological groups in which the inverse mapping \(x\mapsto x^{-1}\) preserves compact subspaces. Section 4 studies Mal'cev spaces and some other algebraic structures on a space; one corollary to the results of this section is Theorem 9 which states: If \(B\) is compact Hausdorff and \(B\) is expressible as the disjoint union of two non-locally compact rectifiable subspaces \(X\) and \(Y\), then either \(X\) and \(Y\) are both pseudocompact or are both Lindelöf \(p\)-spaces.