  The (co)homology self-closeness number of a simply-connected based
CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$
inducing an automorphism of the (co)homology groups for dimensions$\leq k$ is a
self-homotopy equivalence. These two numbers are homotopy invariants and have a
close relation with the group of self-homotopy equivalences. In this paper, we
compare the (co)homology self-closeness numbers of spaces in certain
cofibrations, define the mod $p$ (co)homology self-closeness number of
simply-connected $p$-local spaces with finitely generated homologies and study
some properties of the (mod $p$) (co)homology self-closeness numbers.
