Metrizability and coconnectedness (Q2471006)

A topological space \(X\) is called coconnected if every continuous map \(f:X^2\to X\) depends on at most one coordinate. Solving a problem stated in \textit{J. Sichler} and \textit{V. Trnková} [Topology Appl. 142, No. 1--3, 159--179 (2004; Zbl 1068.54009)], the author constructs metric spaces \(X=(P,\mu)\) and \(Y=(P,\nu)\) such that the four monoids \(\text{Top}(X, X)\), \(\text{Unif}(X, X)\), \(\text{Top}(Y, Y)\) and \(\text{Unif}(Y, Y)\) are formed by the same maps \(P\to P\) and there is a non-expanding map \(X\times X\to X\) depending on both coordinates, but the space \(Y\) is coconnected in Top. This result implies Theorem 2 below and, together with the result by Sichler and Trnková quoted above, also Theorem 1 below. A concrete category \(({\mathcal K}, U)\) with concrete finite products is called CC-comprehensive if there exist objects \(a, b\) in \({\mathcal K}\) such that (i) \(U(a)=U(b)\) and \({\mathcal K}(a, a)\), \({\mathcal K}(b, b)\) determine the same monoid of maps \(U(a)\to U(a)\), and (ii) \(a\) is coconnected in \({\mathcal K}\) but \(b\) is not coconnected in \({\mathcal K}\). Theorem 1. The categories \(M\)Top, \(M\)Unif and Metr are CC-comprehensive, where \(M\)Top (\(M\)Unif) is the full subcategory of Top (Unif) generated by all metrizable spaces, and Metr is the category of metric spaces of diameter at most 1 and all their nonexpanding maps. Let \(({\mathcal K}, U)\) and \(({\mathcal H}, V)\) be concrete categories with concrete finite products, and \(\Phi:{\mathcal K}\to{\mathcal H}\) be a functor which preserves finite products and satisfies \(V\circ\Phi=U\). Then \(\Phi\) is called simultaneously CC-comprehensive if there exist objects \(a, b\) in \({\mathcal K}\) such that (i) \(U(a)=U(b)\) and all the four endomorphism monoids \({\mathcal K}(a, a)\), \({\mathcal K}(b, b)\), \({\mathcal H}(\Phi(a), \Phi(a))\), \({\mathcal H}(\Phi(b), \Phi(b))\) determine the same monoid of morphisms \(U(a)\to U(a)\), and (ii) \(a\) is not coconnected in \({\mathcal K}\) but \(\Phi(b)\) is coconnected in \({\mathcal H}\). Theorem 2. The forgetful functor \(M\text{Unif}\to M\text{Top}\) is simultaneously CC-comprehensive, but neither the forgetful functor \(\text{Metr}\to M\text{Unif}\) nor \(\text{Metr}\to M\text{Top}\) is simultaneously CC-comprehensive.