The embeddability ordering of topological spaces

For K a set of topological spaces and X , Y ∈ K , the notation X ⊆ h Y means that X embeds homeomorphically into Y; and X ∼ Y means X ⊆ h Y ⊆ h X . With X ˜ : = { Y ∈ K : X ∼ Y } , the equivalence relation ∼ on K induces a partial order ⩽ h well-defined on K / ∼ as follows: X ˜ ⩽ h Y ˜ if X ⊆ h Y . sets ( P , ⩽ P ) and ( Q , ⩽ Q ) , the notation ( P , ⩽ P ) ↪ ( Q , ⩽ Q ) means: there is an injection h : P → Q such that p 0 ⩽ P p 1 in P if and only if h ( p 0 ) ⩽ Q h ( p 1 ) 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 ) . thors prove two theorems which improve and extend results from the extensive relevant literature. m 2.2 is a zero-dimensional Hausdorff space S with | S | = κ such that ( P ( S ) / ∼ , ⩽ h ) is a κ-universal poset. em 3.1 are a compact, connected Hausdorff space S and a set K of ( 2 κ -many) compact, connected subspaces of S such that (a) the posets ( P ( κ ) , ⊆ ) and ( K / ∼ , ⩽ h ) are isomorphic; and (b) ( K / ∼ , ⩽ h ) is a κ-universal poset. Further, one may arrange | S | = w ( S ) = | X | = w ( X ) = ℵ κ ⋅ c for each X ∈ K .