Metrizability of rectifiable spaces (Q384153)

A topological space \(G\) is rectifiable if there are continuous maps \(p:G^2\to G\) and \(q:G^2\to G\) such that \[ p(x,q(x,y))=q(x,p(x,y))=y \text{ and } q(x,x)=q(y,y) \] holds for all \(x,y\in G\). A space \(X\) is submaximal if \(A\) is open in \(\bar A\) for every \(A\subseteq X\). A space \(X\) has point-countable type if for every \(x\in X\) there is a compact \(B\subseteq X\) with countable base of neighbourhoods in \(X\) such that \(x\in B\). { Theorem 1.} Let \(G\) be a submaximal rectifiable regular \(T_1\)-space. If \(G\) is locally countably compact or has point-countable type, then \(G\) is metrizable. The paper contains also some other conditions for the metrizability of rectifiable spaces. The following question arizing in the paper seems to be of some interest. { Question 1.} Is every pseudocompact submaximal rectifiable space finite? In this context it should be pointed out that every separable submaximal rectifiable space is countable and even more generally that the density of a submaximal rectifiable space \(G\) is equal to \(card\) \(G\), as is proved in the paper.