How do I check if a file exists on another (remote) domain?

CMW has asked for the wisdom of the Perl Monks concerning the following question:

I am trying to create a script which checks if a site is up or down by checking for a certain file on that domain.

I have tried a few things but as it seems they are all for checking local file and to check for a remote file!

Any suggestions?

Originally posted as a Categorized Question.