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) in U}.
If X has an open universal set parametrised by Y and n in \omega, 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 G\delta-diagonal, then hd(X\omega) <= hL(Y), hL(X\omega) <= hd(Y) and hc( X\omega) <= hc(Y).
The statement "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.
Relevant examples are presented.
Keywords: open universal, cardinal invariants, compact, cometrisable
MSC: 54H05, 54E35, Secondary 54D65, 54D30, 54D15
Date received: September 24, 1999.