My Computer (iMac) Catalina 10.15.7 says the Safari is not open. It will not run, what can I do?

Hello, rjstef.

SAFE MODE

If you haven't rebooted into safe mode anytime recently you should try that to see if it helps. Let the Mac OS do drive directory repair, cache clearing and other misc housekeeping. Often just booting into safe mode and logging in, then immediately rebooting normally will resolve an issue.  You should do this with all your external peripherals disconnected but the keyboard and mouse.

Booting into Safe Mode… 

Use safe mode to isolate issues - Apple Support

When you restart your Mac, press and hold the <shift key> until you see the Apple logo and progress bar.

Note– starting up in Safe mode could take 10 minutes or more. Be patient.

Does the problem persist while in Safe mode? 

Restart your Macintosh and re-evaluate the issue.

I've had the same issue except after installing Big Sur. Yet, in order to install Big Sur, like others, I had to update in Safe Mode. I'll try doing the steps you note for Safari in Safe Mode but makes one wonder, do we have to do this every time we want to use Safare?