Component of a pulled-back universal cover
= Component of a pulled-back universal cover
{title2=$\ker i_*$}
For a locally path-connected subspace $A\subseteq X$, a selected component of the inverse image in a <universal cover> covers the corresponding component $A_0$ of $A$. Its covering <subgroup> is $\ker(\pi_1(A,a_0)\to\pi_1(X,a_0))$; surjectivity onto all of $A$ additionally requires $A$ to be path connected.