  The purpose of this paper is to make a comprehensive connection between the
basic results and properties derived from the two kinds of topologies (namely
the $(\epsilon,\lambda)-$topology introduced by the author and the stronger
locally $L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.
al) for a random locally convex module. First, we give an extremely simple
proof of the known Hahn-Banach extension theorem of $L^{0}-$linear functions as
well as its continuous variants. Then we give the essential relations between
the hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct.
Anal.256(2009)3996--4029] and a basic strict separation theorem in [Guo et. al,
Nonlinear Anal. 71(2009)3794--3804]: in the process obtain a useful and
surprising fact that a random locally convex module with the countable
concatenation property must have the same completeness under the two
topologies! Based on the relation between the two kinds of completeness, we go
on to present the central part of this paper: we prove that most of the
previously established deep results of random conjugate spaces of random normed
modules under the $(\epsilon,\lambda)-$topology are still valid under the
locally $L^{0}-$convex topology, which considerably enriches financial
applications of random normed modules.
