When heating a dilute sample of the binary system of tetraethyleneglycol dodecyl ether (C12E4) and water from the micellar phase (L-1) into the two-phase region of a lamellar phase (L-alpha) and excess water (W) vesicles are formed. During heating, one passes a region of phase separation in the micellar phase (L-1' + L-1") where the initial micelles rapidly fuse into larger aggregates forming the concentrated L-1 phase (L-1") with a structure of branched cylindrical micelles, a so-called "living network". The static correlation length of the micelles are increasing with increasing concentration, from ca. 10 nm to 80 nm in the concentration range of 0.0001 g/cm(3)-0.0035 g/cm(3). The overlap concentration was determined to 0.0035 g/cm(3). When the temperature reaches the L-1' + L-alpha region the network particles transform into bilayer vesicles with a z-average apparent hydrodynamic radius in the order of 200 nm depending on the composition. The size of the final vesicles depends on the extent of aggregation/fusion in the L-1' + L-1" region and hence on the rate of heating. The aggregation/fusion in the L-1' + L-1" is slower than diffusion-limited aggregation, and it is shown that 1/100 of the collisions are sticky results in the fusion event.