Let \(S\) be a quasiordered set (quoset) or topological space and \(\text{Sub} S\) be the set of all nonempty subquosets or subspaces quasiordered by embeddability, respectively. For a cardinal number \(n\), \(p_n\) and \(q_n\) denote the smallest size of spaces \(S\) such that each poset respectively quoset with \(n\) points is embeddable in \(\text{Sub} S\). The authors prove the following results: 1. \(n+1\leq p_n\leq q_n\leq p_n+l(n)+ l(l(n))\), where \(n\) is finite and \(l(n)=\min \{k\in\mathbb{N}: n\leq 2^k\}\). 2. \(n+l(n)-1\leq b_n\leq n+l(n)+l (l(n))+2\), where \(b_n\) is the smallest size of \(S\) such that \(\text{Sub} S\) contains a principal filter isomorphic to the power set \(P(n)\). 3. \(b_n=p_n= q_n=n\), for infinite \(n\).