Spaces in the brain can refer either to psychological spaces, which are derived from similarity judgments, or to neurocognitive spaces, which are based on the activities of neural structures. We want to show how psychological spaces naturally emerge from the underlying neural spaces by dimension reductions that preserve similarity structures and the relevant categorizations. Some neuronal representational formats that may generate the psychological spaces are presented, compared and discussed in relation to the mathematical principles of monotonicity, continuity and convexity. In particular, we discuss the spatial structures involved in the connections between perception and action, for example eye-hand coordination, and argue that spatial organization of information makes such mappings more efficient.