Question

Does i-weight of a zero-dimensional space coincide with zero-dimensional i-weight? Oleg Okunev

Suppose X is a Hausdorff zero-dimensional space with countable i-weight (that is, X admits a continuous bijection onto a Tychonoff space of countable weight). Is there a continuous bijection of X onto a zero-dimensional Tychonoff space of countable weight?

An equivalent formulation: if the family of all continuous functions from X to {0,1} separates points of X, and there is a countable family of continuous real functions on X that separates points of X, must there exist a countable family of continuous functions from X to {0,1} that separates points of X?

The question arises in the theory of spaces of continuous functions to {0,1} with the topology of pointwise convergence: it is easy to prove that if X is zero-dimensional, then C_p(X,2) is separable iff X admits a continuous bijection onto a zero-dimensional Tychonoff space of countable weight; it appears natural to ask whether this is equivalent to the countability of i-weight (which characterizes the separability of C_p(X)).

The answer is ``yes'' for spaces with dimX = 0.

Received by the editors: March 21, 2001