Realizing quasiordered sets by subspaces of `continuum-like' spaces (Q1300335)

Given an ordered set \(E\) and a topological space \(X\), it is said that \(E\) can be realized within \(X\) if there is an injection \(j\) from \(E\) into the class of (homeomorphism classes of) subspaces of \(X\) such that, for \(x,y\) in \(E\), \(x\leq y\) if and only if \(j(x)\) is homeomorphically embeddable into \(j(y)\). If \(X\) is any \(T_3\) space whose nonempty open sets have the same infinite cardinality \(b\) and which has a dense subset of cardinality \(a\) such that \(b^a=b\), then every poset of cardinality \(b\) can be realized within \(X\). Any quasiordered set on \(2^c\) points (or fewer) whose partially ordered skeleton has \(c\) points (or fewer) is realizable in \(R\).