STRONG MEASURE ZERO SETS FILTERS AND POINTWISE CONVERGENCE

Abstract

The notion of a strong measure zero space but not this terminology was introduced in
by E Borel A separable metric space X has strong measure zero if there is for each sequence n n N of positive real numbers a par tition X nXn such that for eac h n the diameter of Xn is less than n In
we found characterizations of the notion of a strong measure zero set in terms of certain selection principles applied to open covers and also in terms of Ramseyan partition relations for certain families of open covers Also the property of having strong measure zero in all nite powers was characterized like this Due to successes in using the concept of a lter on the set of natu ral numbers to describe certain covering properties of sets of real numbers as in
and of using the closure properties of function spaces as in
the question arose whether these methods would also yield analogous descriptions of strong measure zero spaces In the paper
we made abeginning in this direction by showing that for certain function spaces derived from a space X the strong measure zeroness of all nite powers of X is equivalent to certain closure properties of the associated function space The function space asso ciated with X was obtained by rst associating with X a subspace TX of the Alexandro double of the closed unit interval
and by then taking the function space CsTX this space diers from the usual space topologized by the topology of pointwise convergence in that we consider a coarser topology described in terms of a dense discrete subspace consisting of the isolated points of TX