The structure theory of T 5 and related locally compact , locally connected spaces under the PFA and PFA ( S ) [ S ]

A detailed structure theorem is shown for locally compact, locally connected, hereditarily normal spaces and for normal, locally compact, locally connected, hereditarily ω1-scwH spaces in models of PFA(S)[S], and for the latter kinds of spaces in models of PFA. Corollaries include a powerful refinement theorem like that for monotonically normal spaces, and the corollary that the spaces involved are [hereditarily] collectionwise normal and [hereditarily] countably paracompact. Among the problems left unsolved and discussed at the end is the ambious question of whether it is consistent that hereditarily normal, locally compact, locally connected spaces are actually monotonically normal. An affirmative solution would also solve the problem of consistency of every perfectly normal, locally compact, locally connected space being metrizable and thus also solve a 1935 problem due to Alexandroff. In [Ny3], the consistency of a strong metrization theorem for manifolds was “proven”: Statement M. Every T5 (i.e., hereditarily normal), hereditarily cwH manifold of dimension > 1 is metrizable. [Here “cwH” stands for “collectionwise Hausdorff” and “manifold” means “connected space in which each point has a neighborhood homeomporphic to R for some positive integer n”; by Invariance of Domain, n is the same for all points.] Also in [Ny3], the consistency of the following more general statement was announced: Statement A. Every (clopen) component of every locally compact, locally connected, T5, hereditarily cwH space is either Lindelöf or has uncountably many cut points. These results rested on a pair of axioms that turned out to be incompatible. The metrization theorem was salvaged [Ny5] within a week after this discovery, but the consistency of Statement A remained in doubt for a decade. But now we have a theorem of which it is an immediate corollary. 1991 Mathematics Subject Classification. (Updated to 2014) Primary: 03E35, 03E75, 54A35, 54D05, 54D15 Secondary: 54D20, 54E35, 54F15, 54F25.