Non-coincidence of dimension functions for metric spaces
by
S. Mrowka
SUNY ay Buffalo

All spaces are metric. In addition to the two usual dimension functions, ind - the small inductive dimension and dim - the covering dimension, we are concerned with one ind_c M = min{ ind [M\tilde] :[M\tilde]  is a completion of M}. The existence of esoteric spaces (i.e., spaces with dim > ind) was known for a long time. It was not known if there are spaces with dim - ind > 1 or spaces with ind_c - ind > 0. We discuss recent consistency results in this direction. Results There is a space \nu\mu₀ with ind = 0 and such that, under a special condition S(\aleph₀) (see below), a) every completion of \nu\mu₀ contains the interval [0, 1], and therefore ind_c \nu\mu = 1; b) every completion of \nu\mu₀² contains the square [0, 1]², and therefore ind_c \nu\mu₀ ² = dim \nu\mu₀ ² = 2; c) \nu\mu₀ ² contains subspaces \gamma such that for every completion [(\gamma)\tilde] of \gamma, we have ind ([(\gamma)\tilde] \\gamma) > 0. S(\aleph₀):If A is a set of cardinality 2^\aleph₀, then the product A^\aleph₀ cannot be written as A^\aleph₀ = F₁ \cup F₂ \cup ..., where each F_n is an F_\sigma-set in the product topology of A^\aleph₀ (A - discrete) and it is countable on all lines parallel to the n-th axis. S(\aleph₀) disagrees with the continuum hypothesis, but it has been shown that it is consistent with ZFC. Problems and possible ways of further attack will also be discusssed.

Date received: February 6, 1998