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The purpose of this paper is to make a comprehensive connection between thebasic results and properties derived from the two kinds of topologies (namelythe $(\epsilon,\lambda)-$topology introduced by the author and the strongerlocally $L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.al) for a random locally convex module. First, we give an extremely simpleproof of the known Hahn-Banach extension theorem of $L^{0}-$linear functions aswell as its continuous variants. Then we give the essential relations betweenthe 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 andsurprising fact that a random locally convex module with the countableconcatenation property must have the same completeness under the twotopologies! Based on the relation between the two kinds of completeness, we goon to present the central part of this paper: we prove that most of thepreviously established deep results of random conjugate spaces of random normedmodules under the $(\epsilon,\lambda)-$topology are still valid under thelocally $L^{0}-$convex topology, which considerably enriches financialapplications of random normed modules.

A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module