Yes, you can specify keep-alive (Although some browsers refuse to do it in an XHR - Opera for one).
However, you still send the full glut of HTTP headers every single request.
True, you still need the headers, what you don't need is a new connection every key stroke. Establishing a new TCP connection is a relatively high cost network item (between all the Acks and Syns you need 3 messages have to go back and forth for it to establish), keep the connection alive, the stuff will run a lot faster.
You can null out the default headers on a XHR so they don't get sent.