The embeddability ordering of topological spaces
by
W. W. Comfort
Wesleyan University
Coauthors: W. D. Gillam

For K a set of topological spaces and X, Y in K, the relation X subset or equal _h Y means: X embeds homeomorphically into Y. If X subset or equal _h Y subset or equal _h X one writes X ~ Y, and [X\tilde]:={Y in K:X ~ Y}. The partial order <= _h is (well-) defined on K/\negthinspace\negthinspace ~ by: [X\tilde] <= _h[Y\tilde] if X subset or equal _h Y.

For posets (A, <= _A) and (B, <= _B), the notation (A, <= _A)\hookrightarrow(B, <= _B) means: there is an injection f from A into B such that a₀ <= _A a₁ in A iff f(a₀) <= _B f(a₁) in B.

The authors simplify some constructions, and extend some results, of Trnková (1986), and of McMaster et al. (1994, 1996, 1999, 2000), as follows. Here \alpha is an infinite cardinal and \beta(\alpha) is the Stone-Cech compactification of the (discrete) space \alpha.

Theorem 1. There is a set S subset or equal \beta(\alpha), with |S|=\alpha⁺, such that every poset (A, <= ) with |A| <= \alpha⁺ satisfies

(A, <= _A)\hookrightarrow(P(\alpha⁺), subset or equal )\hookrightarrow(P(S)/\negthinspace\negthinspace ~ , <= _h).

Theorem 2. There is a compact, connected, Hausdorff space X_\alpha and a family K_\alpha of (2^\alpha-many) compact, connected subspaces of X_\alpha such that the posets (P(\alpha), subset or equal ) and (K_\alpha/\negthinspace\negthinspace ~ , <= _h) are isomorphic; one may arrange that |X|=w(X)=\aleph_\alpha·c for each X in K_\alpha subset or equal P(X_\alpha).

Date received: May 25, 2003