THE EXTRARESOLVABILITY OF SOME FUNCTION SPACES

A space X is said to be extraresolvable if X contains a family 7) of dense subsets such that the intersection of every two elements of 7) is nowhere dense and 17)1> ~(X), where~(X) = min{IUI: U is a nonempty open subset of X} is the dispersion character of X. In this paper, we study the extraresolvability of some function spaces Cp(X) equipped with the pointwise convergence topology."liVeshowthat Cp(X) is not extraresolvable provided that X satisfies one of the following conditions: X is metric; nw(X) =w; X is normal,e(X) =nw(X) andeithere(X) is attained or cf(e(X)) is countable. Hence,Cp(JR) and Cp(Q) arenot extraresolvable."liVestablishtheequivalences2w <2WI iff Cp([O,WI)) is extraresolvable; and, under GCH, for every infinite cardinal K, the space Cp([O, K)) is extraresolvable iff ef(K)> w, where [0, K) has the ordertopology. \Ve also provethat if K<cf(K) =K and cf(K)>w, then Cp({O,I}K) is extraresolvablejand that Cp(,B(K)) is extraresolvable,for everyinfinitecardinalK with thediscretetopology. It is shownthat Cp([O,,BwI)) is extraresolvable,where,BwI is the beth cardinal correspondingto WI ' Under GCH, for a compactspaceX, we havethat cf(w(X))> W iff Cp(X) is extraresolvable.We proved that 2W < 2WI is equivalento the statement"Cp ( {O,1}WI) is strongly extraresolvable". O