Cerebrospinal fluid is filled between:

Subarachnoid space is a wide space between the pia and the arachnoid, filled with cerebrospinal fluid (CSF). It surrounds the brain and spinal cord like a water cushion. The spinal subarachnoid space is wider than the space around the brain. It is widest below the lower end of the spinal cord where it encloses the cauda equina. Lumbar puncture is usually done in the lower widest part of the space, between third and fourth lumbar vertebrae. Reference: B D Chaurasia’s Head and Neck anatomy, 6th ed page no 184