The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation Xh_⊆Y means that X embeds homeomorphically into Y; and X∼Y means Xh_⊆Yh_⊆X. With , the equivalence relation ∼ on K induces a partial order h_? well-defined on K/∼ as follows: if Xh_⊆Y.For posets (P,P_?) and (Q,Q_?), the notation (P,P_?)?(Q,Q_?) means: there is an injection such that p₀P_?p₁ in P if and only if h(p₀)Q_?h(p₁) in Q. For κ an infinite cardinal, a poset (Q,Q_?) is a κ-universal poset if every poset (P,P_?) with |P|?κ satisfies (P,P_?)?(Q,Q_?).The authors prove two theorems which improve and extend results from the extensive relevant literature.

Theorem 2.2. There is a zero-dimensional Hausdorff space S with|S|=κsuch that(P(S)/∼,h_?)is a κ-universal poset.