Announcement-and-challenge covering of a closed payoff

After a fixed even-depth prefix, Player I announces a move and a set $X$ of first exit positions from a closed set of infinite plays. Player II accepts, or challenges an exit $r\in X$. Acceptance follows the old game until the first exit, where I wins exactly when that exit belongs to $X$. A challenge forces play through $r$ and then resumes the old game. Infinite accepted plays remain in the closed payoff; infinite challenged plays are outside it. Thus the pulled-back payoff is clopen. For an I-strategy, transfer by initially accepting and switching to the challenged simulation at an exit in $X$. For an II-strategy, announce the set of exits that this strategy never challenges under any announcement; it must accept this set. An exit outside it permits a challenged simulation, while an exit inside it is a premature loss for II. These maps prove the game covering lifting condition, rather than merely continuity of the projection.