When you call a PHP page through AJAX you are actually making a new request to the server to interpret that page. The server does not know anything about the page being requested through AJAX or by typing the address in the browser bar, it just interprets it as a new PHP page.
That said, all pending connections are closed automatically, you can put mysql_close
if you wish, otherwise PHP will do it for you.
See the manual page for mysql_close
Using mysql_close() isn't usually
necessary, as non-persistent open
links are automatically closed at the
end of the script's execution.