Ric

A "hot" backup or "online" backup is performed is with the database open, and for this to work you MUST be running in archivelog mode

e.g.

alter tablespace users begin backup;
host cp /u01/oradata/prod/users01.dbf /backup
host cp /u02/oradata/prod/users02.dbf /backup
alter tablespace users end backup;
...
alter system archive log current;

In addition you must also backup your controlfile. You'll probably also want to backup init.ora/spfile, archivelogs, and the password file if you use one.

On the other hand, a "cold" or "offline" backup involves shutting down the database (normal, immediate, or transactional) and making a copy of all the datafiles/init.ora etc. An offline backup is not related to the command to take a tablespace offline.

If you choose to implement offline backups you can and should still run the database in archivelog mode. You can still perform complete recovery using files backed up via an offline backup by applying all archive/online redo.