Local Dimension of Normal Spaces

Introduction LET dim-X be the covering dimension of a space X and let ind X and Ind X be the dimensions defined inductively in terms of the boundaries of neighbourhoods of points and closed sets respectively. The local dimension loo dim X is the least number n such that every point has a closed neighbourhood U with dim U ^ n. The local inductive dimension locIndX is defined analogously, while indX is already a local property. The subset theorem, that dim A < dim X for A c X, which was proved by E. Cech (3) for perfectly normal spaces is here extended to totally normal spaces. Cech's problem (4) of whether the subset theorem holds for completely normal Hausdorff spaces is reduced to the problem of whether the local dimension of a completely normal Hausdorff space is always equal to its dimension: that is, whether locdimX = dimX for X completely normal and Hausdorff. But it follows from [3.7] below that a completely normal space X such that locdimX < dimJf, if any such exists, must be neither paracompact nor the union of a sequence of closed paracompact sets nor the union of two paracompact sets one of which is closed. Thus most of the usual methods of constructing counter-examples are excluded. However, a normal space M is constructed for which