The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation X⊆hY means that X embeds homeomorphically into Y; and X∼Y means X⊆hY⊆hX. With , the equivalence relation ∼ on K induces a partial order ⩽h well-defined on K/∼ as follows: if X⊆hY.For posets (P,⩽P) and (Q,⩽Q), the notation (P,⩽P)↪(Q,⩽Q) means: there is an injection such that p0⩽Pp1 in P if and only if h(p0)⩽Qh(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|=κ such that (P(S)/∼,⩽h) is a κ-universal poset. Theorem 3.1 – There are a compact, connected Hausdorff space S and a set K of (κ2-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)=ℵκ⋅c for each X∈K.