  We are presenting proofs of fundamental results related to homotopy
idempotents, proofs that are sufficiently simple so that even the author can
understand them. The first one is that homotopy idempotents in the category of
pointed connected CW complexes split and the second one is that unpointed
homotopy idempotents in the category of finite-dimensional CW complexes split.
Some of our proofs rectify gaps in the existing literature.
