The concept of
${{C}_{k}}$
-spaces is introduced, situated at an intermediate stage between
$H$
-spaces and
$T$
-spaces. The
${{C}_{k}}$
-space corresponds to the
$k$
-th Milnor–Stasheff filtration on spaces. It is proved that a space
$X$
is a
${{C}_{k}}$
-space if and only if the Gottlieb set
$G(Z,\,X)\,=\,[Z,\,X]$
for any space
$Z$
with cat
$Z\,\le \,k$
, which generalizes the fact that
$X$
is a
$T$
-space if and only if
$G(\sum B,\,X)\,=\,[\sum B,\,X]$
for any space
$B$
. Some results on the
${{C}_{k}}$
-space are generalized to the
$C_{k}^{f}$
-space for a map
$f\,:\,A\,\to \,X$
. Projective spaces, lens spaces and spaces with a few cells are studied as examples of
${{C}_{k}}$
-spaces, and non-
${{C}_{k}}$
-spaces.