No. The connection made by fsockopen is closed when the script ends.
You could check out
pfsockopen. It works exactly like
fsockopen, except that the connection is kept open after the PHP script ends. Which - in theory - allows you to re-connect to the same resource by calling the function again in a new request, using the same parameters. - It's a bit more complicated though, and it may not work as expected on some web-servers.
You could always try do create a background script to manage the resources, that you would call from your web-scripts, but that's an entirely different problem.