6 · Try to choose a continuous section

A cover gives possible points above each point of a probe. A section chooses one point above each point, and those choices must vary continuously.

Try this. Alternate between the two choices above the approaching sequence. No choice at the limit can make that section continuous. Then switch to the extremally disconnected case, where every cover has some continuous section.

Definition 2.4 calls a compact Hausdorff space extremally disconnected when every surjection onto it splits. Warning 2.6 says that a convergent sequence in such a space must eventually be constant.