The brain has a large system of cavities called ventricles. This system is called the ventricular system. The cavities do not contain any neural matter, but they are not empty. The ventricles produce a fluid called cerebro-spinal fluid (CSF). This fluid is critical for delivering nutrients to both grey and white matter. Cerebro-spinal fluid bathes both the brain and the spinal cord.
The corpus callosum is a large white matter structure within the brain that spans and connects the left and the right hemispheres of the cerebrum. It is located anterior to the lateral ventricles. The lateral ventricles are the largest ventricles in the brain. Each hemisphere of the cerebrum has a lateral ventricle, so with regard to the corpus callosum, there are actually two ventricles next to it: the left and right lateral ventricles.