Since the IOS Ventura 13.5 download this week I no longer receive or can send emails via my iMac

First thing to try before diving in too deep, is to boot your computer into safemode. This process runs its own diagnostics and will look for and fix many problems. After that, reboot back into your normal login and test mail again.

Start up your Mac in safe mode - Apple Support

The initial trouble shooting on this type of issue from Apple Support Document may help

 If you can’t send or receive email on your Mac 

Though, it would really help to know Who your e-mail account is with ? Yahoo, G-Mail, Outlook, iCloud etc

A glitch like this " Off Line " can occur but generally is not caused by macOS 13.5 updates.

At least not on 3 different machines I own and were running Ventura 13.4.1(c) ( previous version ) and then updated to 13.5

More importantly, you have discovered the problem and it works as expected once again

The Original Issue, as you have previously outlined " Sorry, it's iCloud. " was the sole focus of the issue

Branching off to include G-Mail, well, will not touch that one.

The two " Elephants in the Room " Google Services" and Apple do not always play well with each other.

Things seem to be fixed, though I can’t say what steps I took that may have fixed it. I deleted and replaced the Mail app on my iPhone which had no effect and I attempted to delete my account and add it back in but that, too, didn’t seem to work either. I could see Mail was online. I was testing my email by sending myself emails and for quite awhile neither the sent emails nor the received emails were showing up, but that’s not surprising, given that nothing was coming or going out.

The short of it: it seems to be working but Incan’t say why?

If you have fixed the iCloud e-mail issue great.

But if your issue remains got G-mail

Sorry but I done not use that service for the reasons already presented.

In which case I am not a good fit to help you with anything related to Google outside of using their Search Engine

This did not work for me. I don't see this issue being addressed anywhere else or any possible solutions?

Thanks for the link! Sorry, it's iCloud.

Then as you are following the Authors @ warrior57 question - the same trouble shooting methods from previous link would apply

I was only use Mail, not G-mail. I have a G-mail account that is not active. Thank you for your help!!