Trees, Gleason spaces, and coabsolutes of $\beta {\bf N}\sim {\bf N}$

Scott W. Williams
1982 Transactions of the American Mathematical Society  
For a regular Hausdorff space X, let &( X) denote its absolute, and call two spaces X and Y coabsolute (S-absolute) when &(X) and &(Y) (ß&(X) and ß&(Y)) are homeomorphic. We prove X is S-absolute with a linearly ordered space iff the POSET of proper regular-open sets of X has a cofinal tree; a Moore space is 9-absolute with a linearly ordered space iff it has a dense metrizable subspace; a dyadic space is ë-absolute with a linearly ordered space iff it is separable and metrizable; if X" is a
more » ... ally compact noncompact metric space, then ßX ~ X is coabsolute with a compact linearly ordered space having a dense set of P-points; CH implies but is not implied by "if X is a locally compact noncompact space of w-weight at most 2" and with a compatible complete uniformity, then ß X ~ X and ßN ~ N are coabsolute." A tree T is a POSET (partially ordered set) in which ]<-,/[, the set of predecessors of /, is well ordered for each / G T. The trees most familiar to topologists are the Cantor tree, the Souslin trees, and the Aronszajn trees [Ku], [Ru]. In §1 we study conditions under which a given POSET contains a cofinal tree. Recall [Po], [P.S.] that if A' is a space,1 then the absolute S( X) of X is the unique (up to a homeomorphism) extremally disconnected space that can be mapped irreducibly onto A" by a perfect map. Following [C.N.2] call ß&( X) the Gleason space of X and denote it by @(X). Two spaces X and Y are coabsolute (^-absolute) whenever &(X) and &(Y) (respectively, §(X) and S(Y)) are homeomorphic. Designate ^l(X) for the Boolean algebra of regular-open sets of A-then it is known that @(X) s §(Y) iff <3L(X) = ft(y). In §2, we begin an application of §1 to topology with several theorems. We prove: (2.1) Ais S-absolute with a linearly ordered space if, and only if, (<3l(X) ~ {X}, D ) contains a cofinal tree. (2.3) ((2.8)) A first countable (Moore) space is S-absolute with a linearly ordered space iff it has a dense linearly ordered (metrizable) subspace. (2.10) A dyadic space is S-absolute with a linearly ordered space iff it is separable and metrizable. We also give (2.6 and 2.7) sufficient conditions (dependent on certain cardinal functions) for a space X to have a dense linearly ordered subspace.
doi:10.1090/s0002-9947-1982-0648079-x fatcat:bhbju2ff5batnko74vivmoomgu