David,
I just checked:
Sendmail was not running as a service.I had deselected it,for some reason.
Do you think this could be the cause of the problem?

Ok, I did the tests as you said (thanks for the idiot -proof language. I appreciate it) and the mail was sent and recieved ok.
Also the
telnet localhost 25 and

telnet localhost smtp worked too.

So I think it was the fact that I didn't have sendmail running.
Would you agree?

no doubt, this was the problem

David,
I'd just like to thank you for helping me and explaining things in a clear logical manner.

I want you to know I really appreciate your help.
Thanks,
Ludwig

No problem, that's why we're here.

For completeness sake, you don't need sendmail running as a service to be able to fetch mail from a remote POP3 account and deliver mail to local users.

"man fetchmail" says: "If no port 25 listener is available, but your fetchmail configuration was told about a reliable local MDA, it will use that MDA for local delivery instead." meaning you will only need running sendmail when you where somehow unable to generate a proper fetchmail configuration file.

Running any network service (out of the box) without proper configuration should be advised against.

here is some info on using an MDA using -m

-m <command>, --mda <command>
(Keyword: mda) You can force mail to be passed to
an MDA directly (rather than forwarded to port 25)
with the -mda or -m option. Be aware that this
disables some valuable resource-exhaustion checks
and error handling provided by SMTP listeners; it's
not a good idea unless running an SMTP listener is
impossible. If fetchmail is running as root, it
sets its userid to that of the target user while
delivering mail through an MDA. Some possible MDAs
are "/usr/sbin/sendmail -oem -f %F %T",
"/usr/bin/deliver" and "/usr/bin/procmail -d %T"
(but the latter is usually redundant as it's what
SMTP listeners usually forward to). Local delivery
addresses will be inserted into the MDA command
wherever you place a %T; the mail message's From
address will be inserted where you place an %F. Do
not use an MDA invocation like "sendmail -oem -t"
that dispatches on the contents of To/Cc/Bcc, it
will create mail loops and bring the just wrath of
many postmasters down upon your head.