Embedding structures

For any quasiordered set ('quoset') or topological space S, the set Sub S of all nonempty subquosets or subspaces is quasiordered by embeddability. Given any cardinal number n, denote by p_n and q_n the smallest size of spaces S such that each poset, respectively, quoset with n points is embeddable in Sub S. For finite n, we prove the inequalities n + 1 ≤ p_n ≤ q_n ≤ p_n + l(n) + l(l(n)), where l(n) = min{k ⊂ ℕ | n ≤ 2^k}. For the smallest size b_n of spaces S so that Sub S contains a principal filter isomorphic to the power set script P sign(n), we show n + l(n) - 1 ≤ b_n ≤ n + l(n) + l(l(n)) + 2. Since p_n ≤ b_n, we thus improve recent results of McCluskey and McMaster who obtained p_n ≤ n². For infinite n, we obtain the equation b_n = p_n = q_n = n.