Connectivity of a CW pair

Suppose the component and relative-triviality conditions hold. A map of a zero-disk is a point of $X$, whose component meets $A$, so a path to $A$ supplies its compression. For $i\ge 1$, mark $b=(1,0,\dots ,0)\in S^{i-1}$ and set $a=u(b)$ for the specified disk map $u$. Its relative class based at this actual $a$ is trivial by hypothesis. By [F3] it has a homotopy into $A$ fixing the full boundary, including both endpoints when $i=1$. No selected family of basepoints or paths is involved.

Conversely suppose all the stated disk-compression conditions hold. The zero-disk condition supplies a path to $A$ for any point of $X$, so every component meets $A$. For each $a\in A$, every positive relative class in degrees $i\le k$ has a disk representative with marked boundary value $a$ by [F3]. Its stipulated boundary-fixed compression makes the class trivial by the converse in [F3]. Thus all the defining conditions hold.

If $A$ is empty and $X$ nonempty, its point disks fail the required path condition; if both are empty there are no such maps or basepoints. When $X=A$, the constant homotopy of any disk map already ends in $A$, so the compression condition holds in all degrees. For $k=0$ there are no positive-degree conditions; for $k=1$ the positive condition is exactly the pointed degree-one one. Steps 1.1–1.2 prove both formulations equivalent, including their endpoints, without any choice principle.