  We reformulate the inequalities among self-closeness numbers of spaces in
cofibrations making use of homology dimension and show that the self-closeness
number of a space is less than or equal to the homology dimension of the space.
Then we prove a relation of self-closeness numbers and the connectivity for
manifolds satisfying Poincar\'{e} duality. On the other hand we determine the
self-closeness numbers of the real projective spaces, lens spaces and a cell
complex defined by Mimura and Toda. Moreover, making use of the models of
Sullivan and Quillen, we show several properties of self-closeness number for
finite cell complexes, and rational examples are udied to obtain some precise
results. Finally, we prove relations among self-closeness numbers defined by
homotopy groups, homology groups and cohomology groups.
