Eureka...eureka... Mosquitto works....
Thanks to your contributions I have achieved what I wanted step by step.... for which I thank you.
Indeed Mosquitto has worked in the beginning but I am expecting an opening window.
Meanwhile, I have successfully send a message from the terminal window 1 to 2.
Thanks to all who have taken the trouble to look at my question and send me a possible solution.
See you later.
JackDG

I just uninstall mosquitto... with sudo apt-get purge --remove mosquitto*

The first time i install with :
sudo wget http://repo.mosquitto.org/debian/mosquitto-repo.gpg.key
sudo apt-key add mosquitto-repo.gpg.key
cd /etc/apt/sources.list.d/
sudo wget http://repo.mosquitto.org/debian/mosquitto-stretch.list
sudo apt-get update
sudo apt-get install mosquitto

I look and study your link.

Tanks Kimot,

I do what you says and this what happen not more.

pi@raspberrypi:~ $ sudo service mosquitto stop
pi@raspberrypi:~ $ sudo systemctl stop mosquitto.service
pi@raspberrypi:~ $ mosquitto -v
1559686190: mosquitto version 1.6.2 starting
1559686190: Using default config.
1559686190: Opening ipv4 listen socket on port 1883.
1559686190: Opening ipv6 listen socket on port 1883.

Thank you for your efforts.
Apparently it doesn't matter that the port is already in use.

Can this help you?
https://randomnerdtutorials.com/how-to-install-mosquitto-broker-on-raspberry-pi/

I can't find a forum for Mosquitto users either.

Greetings.

JackDG

I have successfully installed Mosquitto on a Raspberry Pi 3 B+ (Debian = stretch).

Here is the result on the Raspberry :

     pi@raspberrypi:~ $ mosquitto -v
     1559603778: mosquitto version 1.6.2 starting
     1559603778: Using default config.
     1559603778: Opening ipv4 listen socket on port 1883.
     1559603778: Error: Address already in use
     pi@raspberrypi:~ $ 

Apparently runs the program but no working window appears on the screen.

What am I doing wrong?

Jack DG