Cofinal ${\displaystyle {\Sigma }^1\_1}$ and ${\displaystyle {\Pi }^1\_1}$ subsets of ${\displaystyle {\omega }^{\omega }}$

We study properties of ${\displaystyle {\sum }^1\_1}$ and ${\displaystyle {\pi }^1\_1}$ subsets of ${\displaystyle {\omega }^{\omega }}$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement "Any compact covering mapping from a Borel space onto a Polish space is inductively perfect" is equivalent to the statement "${\displaystyle \forall \alpha \in {\omega }^{\omega },{\omega }^{\omega }\cap L(\alpha )}$ is bounded for ≤*".