Open universal sets

All spaces are assumed to be regular Hausdorff topological spaces. Let X and Y be spaces. An open subset U of X×Y is said to be an open universal set for X parametrised by Y if for all open V in X there is an element y of Y such that V={x: (x,y)∈U}. as an open universal set parametrised by Y and n∈ω, then w(X)⩽nw(Y), hd(Xn)⩽hL(Yn), hL(Xn)⩽hd(Yn) and hc(Xn)⩽hc(Yn). If X is also compact, then hL(Xn)⩽hL(Yn) and hd(Xn)⩽hd(Yn). If X has a Gδ-diagonal, then hd(Xω)⩽hL(Y), hL(Xω)⩽hd(Y) and hc(Xω)⩽hc(Y). atement ‘every compact zero-dimensional space with an open universal set parametrised by a space with the hereditary c.c.c. is metrisable’ is consistent and independent of ZFC. The statement ‘every cometrisable space with an open universal parametrised by a hereditarily c.c.c. space is metrisable’ is consistent and independent. nt examples are presented.