The Tukey order on compact subsets of separable metric spaces (Q2805030)

The Tukey order of partial orders is defined by \(P\geq_TQ\) if there is a map \(f:P\to Q\) that maps cofinal sets to cofinal sets (equivalently: some \(g:Q\to P\) maps unbounded sets to unbounded sets).NEWLINENEWLINEThe authors define a relative version for pairs~\((P',P)\) where \(P'\)~is a subset of the partial order~\(P\): \((P',P)\geq_T(Q',Q)\) if there is \(f:P\to Q\) such that if \(C\subseteq P\) satisfies \((\forall p\in P')(\exists c\in C)(p\leq c)\) then \(f[C]\)~satisfies \((\forall q\in Q')(\exists c\in C)\bigl(q\leq f(c)\bigr)\). The main example of interest is \(\bigl(X,\mathcal{K}(X)\bigr)\), where \(X\)~is a topological space and \(\mathcal{K}(X)\) the family of compact subsets (\(X\)~is identified with the family of one-point sets). One of the main applications is to the class \(\bigl\{\mathcal{K}(X):X\)~is separable metric\(\bigr\}\): it has a chain of length \(\mathfrak{c}^+\) and an antichain of cardinality~\(2^{\mathfrak c}\).