We studied the paranematic ordering induced by a polymer network in the isotropic phase of a liquid crystal (LC) that occurs in polymer-stabilized cells with bend configuration of the LC director (π cells) fabricated via photopolymerization of photoreactive monomer RM 82 added in small concentrations (3-5 wt %) to a nematic LC [4-cyano-4'-pentylbiphenyl (5CB)] when low voltage was applied across the cell. The polymer network formed in the nematic phase of the LC consists of fine fibrils that are aligned along the LC director and thus mirror the bend deformation of the LC at the time of polymerization. When heated to temperatures above the nematic-to-isotropic (N-I) phase transition such highly ordered polymer network anchors LC molecules providing ordering of the LC around the fibrils which results in unusually high optical retardation of the cell, R_{cell}. We present a theoretical model that relates R_{cell} to the degree of order of the fibrils, the anchoring energy of the LC molecules on the surface of the polymer fibrils, and the fibril radius r_{0}. Fitting of the experimental R_{cell}(T) curves with the developed model reveals correlation of r_{0} with the nematic correlation length ξ_{0} which characterizes penetration of the nematic order in the isotropic phase of the LC. Accepting ξ_{0} as a material constant of about 1 nm leads to a very small radius of the fibrils, r_{0}∼1nm, which is also supported by other reported experimental data. High optical retardation and fast electro-optical response of the cells at the temperatures deep into the isotropic phase point toward the enhancement of the polymer-induced paranematic order by a well-oriented layer of LC molecules that are absorbed on the surface of fibrils. Application of high voltage at the isotropic phase temperatures results in high variations of the optical retardation of the cells. Characteristic on and off response times were about 10-100 μs, independent of the cell gap. Combination of large voltage-driven changes of the optical retardation occurring in the low-viscosity isotropic state with switching times that are at least two orders of magnitude shorter than the typical relaxation times of the cells operating in the nematic phase make such polymer-stabilized π cells very promising for application in fast electro-optical switches and light modulators.