Let
C
c
(
T
)
{C_c}(T)
denote the space of continuous real-valued functions on a completely regular Hausdorff space T, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which
C
c
(
T
)
{C_c}(T)
is bornological, barrelled, and infrabarrelled. In this paper the question of when
C
c
(
T
)
{C_c}(T)
is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for
C
c
(
T
)
{C_c}(T)
to have CSMP is obtained.