If your account is IMAP then this is the expected behavior.

Second part of nancymail post. First, please bear in mind I am a 75year old grandmother who does not understand sophisticated technology.

Background info - We have 4 PCs: 2 main desktops, a laptop and a netbook. We do not own smartphones so we are not sync with a cell phone.

Just check Thunderbird "Server settings" and here is what we learned: The 2 desktops and the netbook are IMAP mail servers Port 993; the laptop is POP3 mail server - Port 995.

I was incorrect -- It was the netbook [NOT the laptop] that when I deleted inbox and sent messages all messages were deleted on the desktops PCs.

If you really want to be able to see any of your email messages on any of these four machines, you need all of them set up to use IMAP, and accept that whatever you do on one machine will be reflected on all the others.

I would expect that any message you view on the laptop (using POP) will disappear from all the other computers, UNLESS you have set this specific machine to "leave a copy on the server".

The messages stored on the POP-connected laptop are yours to keep; they are stored locally and you can remove them from the account on any of the other computers and they will stay in place on this laptop. For some users this is a valuable attribute of the POP system. The huge disadvantage of POP, IMHO, is that if you file your messages into separate folders on the other computers, this one laptop won't be able to see those other folders.

If you use IMAP on all your computers, then if you want to tidy up an account, either move messages into subfolders, so they are out of sight but close to hand, or if you're approaching capacity limits of the account's server, move them to Local Folders on whichever machine you elect to be the archive keeper.