  For the study of some typical problems in finance and economics,
\v{Z}itkovi\'{c} %[G. \v{Z}itkovi\'{c}, Convex compactness and its
applications, Math. Finan. Eco., 3(1)(2010) 1--12] introduced convex
compactness and gave many remarkable applications. Recently, motivated by
random convex optimization and random variational inequalities, Guo, et al
introduced $L^0$-convex compactness, developed the related theory of
$L^0$-convex compactness in random normed modules and further applied it to
backward stochastic equations. %[T.X. Guo, et al, Two fixed point theorems in
complete random normed modules and their applications to backward stochastic
equations, J. Math. Anal. Appl., 483(2020) 123644]. In this paper, we
extensively study $L^0$-convexly compact sets in random locally convex modules
so that a series of fundamental results are obtained. First, we show that every
$L^0$-convexly compact set is complete (hence is also closed and has the
countable concatenation property). Then, we prove that any $L^0$-convexly
compact set is linearly homeomorphic to a weakly compact subset of some locally
convex space, and simultaneously establish the equivalence between $L^0$-convex
compactness and convex compactness for a closed $L^0$-convex set. Finally, we
establish Tychonoff type, James type and Banach-Alaoglu type theorems for
$L^0$-convex compactness, respectively.
