The Arabian Sea (known in Latin as the Mare Arabicum et Indicum meaning the Arabian and Indian Sea) was a sea on planet Earth. This sea, which was the northwestern portion of the Indian Ocean, was located between India and the Arabian Peninsula. This sea was located north of the equator.
Maps of Earth were stored in the USS Enterprise's library computer in 2254. In a relief map, the Arabian Sea was seen on this map. It was among the materials viewed by the Talosians when they scanned the Enterprise computer. (TOS-R: "The Cage")
The map from the Enterprise opening sequence, known as "Harmonia macrocosmica seu atlas universalis et novus, totius universi creati cosmographiam generalem, et novam exhibens", was created by Andreas Cellarius. A scan of the original can be found here.
This sea was labeled on a 21st century-era political map of Earth seen in the remastered "The Cage". This was a Central Intelligence Agency map published either in April 2007 or September 2008.   The difference, which was not noticeable at the resolution of the image from the episode, was the addition of Kosovo, which gained its independence between the releases. The label "Arabian Sea" was not legible to the viewer.
Retroactive continuity: A 20th century-era map of western and southern Asia and northern and eastern Africa was stored in the USS Enterprise library computer in 2254. The Arabian Sea was labeled on this map. This map was among the materials viewed by the Talosians when they scanned the Enterprise computer. (TOS: "The Cage")