In our prior study the negative correlation between Topology, a behavioral measure of fluid reasoning, and adult age diminished with the increase in the level of expertise in a cognitively-demanding domain of expertise in the game of GO. The present fMRI study was designed to investigate neural substrates of Topology. The modified topology sub-test of Cattell's Culture Fair Intelligence Tests was used as cognitive stimuli. The results indicated that higher-order cognition of Topology was supported by neural networks in: (1) the parietal cortex which is involved in activating possible responses based on learned stimulus-response associations, and (2) the prefrontal cortex that is recruited when there is a need to generate and evaluate hypotheses, and select between competing responses. Our results were consistent with previous neuroimaging studies of reasoning using Raven's Progressive Matrices that revealed the engagement of the prefrontal cortex and the parietal cortex in inductive reasoning. The future need was discussed to systematically examine neural networks supporting Topology over the course of expertise development and adult development in order to specify unique aspects of fluid reasoning that support high levels of cognitive behaviors of Topology. (Contains 2 figures and 1 table.)