"check out" = "open"
"check in" = "submit"
P4V uses its own terminology that's sort of a mishmash of Perforce and VSS terms with a couple of unique additions sprinkled in, which is why it replaces the Perforce term "open for edit" with "check out", and "sync" with "get", but keeps "submit" as "submit", and also replaces "open for add" with "mark for add" because why not? Regardless, "submit" is the "check in" command you're looking for.
After you submit a file, it should no longer be open (unless you reopen it -- there's an option to automatically reopen files after submit).
In addition, by default other users should be able to open files that you have open (the exception is if it has the "+l" aka "exclusive open" filetype).
So if you have both the "reopen after submit" option ticked in your client AND the "+l" filetype on all your files, you should turn off at least one of those options -- but neither of those is the default so I'm guessing something else is going on. Maybe if you posted an example of what you're talking about?